SAT Math Advanced Math practice: 10 worksheets with answer keys

Practice for the Advanced Math domain of the digital SAT Math section: quadratics and parabolas, equivalent expressions and polynomials, exponential functions and nonlinear systems. Each worksheet starts easy and gets harder, most open with worked examples and an SAT shortcut, and every one has an answer key with brief explanations.

The stretch sheets at the end go a little beyond the SAT, into IB Higher Level ideas such as complex roots, vectors and logarithmic models: good practice for a strong student, but not needed for a top SAT score. Every question is our own: none is from College Board.

Worksheets on this page

Notes for quadratics: Nonlinear functions.

Quadratics and parabolas

Quadratics · 20 questions

Worked Examples: SAT Strategy

Example 1: Easy (Bypassing Calculus for Max/Min)

Question: The function $h(t) = -16t^2 + 64t + 80$ models the height of a projectile in feet $t$ seconds after launch. What is the maximum height reached by the projectile?
SAT Strategy: Instead of finding $h'(t) = 0$, use the vertex formula $x = -\frac{b}{2a}$ to find the time of the maximum, then substitute it back in.
  1. Identify $a = -16$ and $b = 64$.
  2. Find the $x$-coordinate (time) of the vertex: $t = -\frac{64}{2(-16)} = -\frac{64}{-32} = 2$.
  3. Substitute $t = 2$ back into the height equation: $h(2) = -16(2)^2 + 64(2) + 80$.
  4. $h(2) = -64 + 128 + 80 = 144$.
  5. Answer: $144$.

Example 2: Medium (The Discriminant with Unknowns)

Question: The equation $3x^2 - 12x + c = 0$ has exactly one real solution. What is the value of the constant $c$?
SAT Strategy: "Exactly one real solution" means the parabola's vertex rests perfectly on the $x$-axis. Mathematically, this means the discriminant ($\Delta = b^2 - 4ac$) must equal exactly zero.
  1. Identify $a = 3$, $b = -12$, and $c = c$.
  2. Set up the discriminant equation: $(-12)^2 - 4(3)(c) = 0$.
  3. Simplify: $144 - 12c = 0$.
  4. Solve for $c$: $12c = 144 \implies c = 12$.
  5. Answer: $12$.

Example 3: Hard (Sum and Product of Roots Shortcuts)

Question: Let $x_1$ and $x_2$ be the solutions to the equation $2x^2 - 9x + 5 = 0$. What is the value of $x_1 + x_2$?
SAT Strategy: Do not use the quadratic formula to find the messy roots just to add them together! The SAT tests whether you know the "Sum of Roots" shortcut: $x_1 + x_2 = -\frac{b}{a}$. (Note: The "Product of Roots" is $\frac{c}{a}$).
  1. Identify $a = 2$ and $b = -9$.
  2. Apply the sum of roots formula: $-\frac{-9}{2}$.
  3. Simplify: $\frac{9}{2}$ or $4.5$.
  4. Answer: $4.5$.

Practice Questions

Level 1: Easy (Foundational SAT)

  1. The function $f(x) = (x - 4)^2 + 7$ is graphed in the $xy$-plane. What is the minimum value of $f(x)$? (Grid-in)
  2. Which of the following equivalent forms of the equation $f(x) = x^2 - 8x + 15$ displays the $x$-intercepts of the parabola as constants or coefficients?
    • A) $f(x) = x^2 - 8x + 15$
    • B) $f(x) = x(x - 8) + 15$
    • C) $f(x) = (x - 4)^2 - 1$
    • D) $f(x) = (x - 3)(x - 5)$
  3. What is the $y$-intercept of the graph of $y = 3(x - 2)(x + 5)$ in the $xy$-plane?
    • A) $(0, -30)$
    • B) $(0, -10)$
    • C) $(0, 3)$
    • D) $(0, 15)$
  4. The equation $x^2 - 6x + k = 0$ has exactly one real solution. What is the value of $k$? (Grid-in)
  5. The graph of $y = -2x^2 + 16x - 5$ in the $xy$-plane is a parabola. What is the $x$-coordinate of the vertex of this parabola? (Grid-in)

Level 2: Medium (Standard SAT)

  1. Let $x_1$ and $x_2$ be the solutions to the equation $3x^2 - 15x + 8 = 0$. What is the value of $x_1 + x_2$?
    • A) $-5$
    • B) $\frac{8}{3}$
    • C) $5$
    • D) $15$
  2. A toy rocket is launched from the ground. Its height $h$, in meters, $t$ seconds after launch is given by $h(t) = -5t^2 + 30t$. How many seconds after launch does the rocket reach its maximum height? (Grid-in)
  3. The quadratic equation $2x^2 - px + 18 = 0$ has exactly one real solution. If $p > 0$, what is the value of $p$? (Grid-in)
  4. Which of the following equations represents a parabola that opens downward and has a vertex at $(-3, 5)$?
    • A) $y = 2(x - 3)^2 + 5$
    • B) $y = -2(x + 3)^2 + 5$
    • C) $y = -2(x - 3)^2 + 5$
    • D) $y = 2(x + 3)^2 - 5$
  5. The equation $y = x^2 - 10x + c$ represents a parabola in the $xy$-plane. If the vertex of the parabola lies on the $x$-axis, what is the value of $c$? (Grid-in)

Level 3: Hard (Advanced SAT)

  1. The function $f$ is defined by $f(x) = (x - p)(x - q)$, where $p$ and $q$ are constants. The graph of $f(x)$ has $x$-intercepts at $(-2, 0)$ and $(8, 0)$. What is the $x$-coordinate of the vertex of the graph?
    • A) $3$
    • B) $4$
    • C) $5$
    • D) $6$
  2. What is the product of the solutions to the equation $4x^2 - 12x - 7 = 0$?
    • A) $-\frac{7}{4}$
    • B) $-3$
    • C) $\frac{7}{4}$
    • D) $3$
  3. The graph of $y = a(x - h)^2 + k$ passes through the point $(2, 14)$ and has its vertex at $(4, 2)$. What is the value of $a$? (Grid-in)
  4. The system of equations $y = x^2 - 6x + 8$ and $y = 8$ intersects at two points in the $xy$-plane. What is the distance between these two points of intersection?
    • A) $4$
    • B) $6$
    • C) $8$
    • D) $10$
  5. The quadratic function $g(x) = -x^2 + bx + c$ has roots at $x = -1$ and $x = 5$. What is the maximum value of $g(x)$? (Grid-in)

Level 4: Very Hard (Slightly beyond SAT)

  1. The function $f(x) = x^2 + 8x + 20$ can be rewritten in the form $f(x) = (x + h)^2 + k$. What is the value of $h + k$? (Grid-in)
  2. A parabola with equation $y = ax^2 + bx + c$ has a vertex at $(2, 9)$ and intersects the $y$-axis at $(0, -3)$. What is the value of $a + b + c$? (Grid-in)
  3. The function $f(x) = a(x - p)(x - q)$ has a maximum value of $36$. If the $x$-intercepts of the graph of $f(x)$ are $x = 2$ and $x = 8$, what is the value of $a$?
    • A) $-4$
    • B) $-2$
    • C) $2$
    • D) $4$
  4. The equation $kx^2 - 6x + k = 0$ has no real solutions. Which of the following best describes the possible values for the constant $k$?
    • A) $k > 3$
    • B) $-3 < k < 3$
    • C) $k < -3$ or $k > 3$
    • D) $k = 3$
  5. The parabola $y = (x - 3)^2 - 16$ intersects the $x$-axis at points $A$ and $B$. If point $V$ is the vertex of the parabola, what is the area of triangle $ABV$? (Grid-in)

Answer key

The answer to every question, with a brief explanation. Try the sheet first.

Quadratic equations

Quadratics · 20 questions

Worked Examples: SAT Strategy

Example 1: Easy (Factoring and Roots)

Question: What is the positive solution to the given equation $x^2 - 8x - 20 = 0$?
SAT Strategy: When $a=1$, immediately look for two numbers that multiply to the constant ($c = -20$) and add to the middle coefficient ($b = -8$).
  1. The factors of $-20$ that add to $-8$ are $-10$ and $+2$.
  2. Rewrite the equation in factored form: $(x - 10)(x + 2) = 0$.
  3. Set each factor to zero to find the roots: $x = 10$ and $x = -2$.
  4. The question asks for the positive solution.
  5. Answer: $10$.

Example 2: Medium (Vertex Form & Minimums)

Question: The function $f$ is defined by $f(x) = 2x^2 - 12x + 22$. What is the minimum value of $f(x)$?
SAT Strategy: The minimum or maximum value of a quadratic function is the $y$-coordinate of its vertex. You can find the $x$-coordinate of the vertex using the formula $x = -\frac{b}{2a}$.
  1. Identify $a = 2$ and $b = -12$.
  2. Calculate the $x$-coordinate of the vertex: $x = -\frac{-12}{2(2)} = \frac{12}{4} = 3$.
  3. Substitute $x = 3$ back into the function to find the minimum value ($y$).
  4. $f(3) = 2(3)^2 - 12(3) + 22 = 2(9) - 36 + 22 = 18 - 36 + 22 = 4$.
  5. Answer: $4$.

Example 3: Hard (The Discriminant)

Question: The equation $x^2 - 14x + c = 0$ has exactly one real solution. If $c$ is a constant, what is the value of $c$?
SAT Strategy: A quadratic equation $ax^2 + bx + c = 0$ has exactly one real solution if and only if its discriminant ($\Delta = b^2 - 4ac$) is exactly equal to $0$. (If $>0$, two solutions. If $<0$, no real solutions).
  1. Identify the coefficients: $a = 1$, $b = -14$, and the constant is $c$.
  2. Set up the discriminant equation: $(-14)^2 - 4(1)(c) = 0$.
  3. Simplify: $196 - 4c = 0$.
  4. Add $4c$ to both sides: $196 = 4c$.
  5. Divide by 4: $c = 49$.
  6. Answer: $49$.

Practice Questions

Level 1: Easy (Foundational SAT)

  1. $(x - 4)(x + 9) = 0$. What is the positive solution to the given equation? (Grid-in)
  2. $x^2 - 25 = 0$. What is the positive solution to the given equation? (Grid-in)
  3. Which of the following expressions is equivalent to $x^2 + 7x + 10$?
    • A) $(x + 2)(x + 5)$
    • B) $(x - 2)(x - 5)$
    • C) $(x + 1)(x + 10)$
    • D) $(x + 3.5)^2 + 10$
  4. The function $f$ is defined by $f(x) = (x - 3)^2 + 5$. What are the coordinates of the vertex of the graph of $y = f(x)$ in the $xy$-plane?
    • A) $(-3, 5)$
    • B) $(3, -5)$
    • C) $(3, 5)$
    • D) $(9, 5)$
  5. $x^2 - 6x + 9 = 0$. How many distinct real solutions does the given equation have?
    • A) Zero
    • B) Exactly one
    • C) Exactly two
    • D) Infinitely many

Level 2: Medium (Standard SAT)

  1. $2x^2 - 16x + 30 = 0$. What is the sum of the solutions to the given equation?
    • A) $-8$
    • B) $8$
    • C) $15$
    • D) $30$
  2. The function $f$ is defined by $f(x) = x^2 - 10x + 24$. What is the minimum value of $f(x)$? (Grid-in)
  3. $x^2 + kx + 36 = 0$. In the given equation, $k$ is a positive constant. If the equation has exactly one real solution, what is the value of $k$? (Grid-in)
  4. $3x^2 + 12x - 15 = 0$. What is a positive solution to the given equation? (Grid-in)
  5. Which of the following expressions is equivalent to $2x^2 + 12x + 18$?
    • A) $2(x + 3)^2$
    • B) $2(x - 3)^2$
    • C) $(2x + 3)^2$
    • D) $(x + 6)^2 - 18$

Level 3: Hard (Advanced SAT)

  1. $ax^2 - 14x + 49 = 0$. In the given equation, $a$ is a constant. If the equation has no real solutions, which of the following could be the value of $a$?
    • A) $0$
    • B) $0.5$
    • C) $1$
    • D) $2$
  2. A parabola in the $xy$-plane intersects the $x$-axis at $(-2, 0)$ and $(6, 0)$ and passes through the point $(0, -12)$. What is the $y$-coordinate of the vertex of this parabola? (Grid-in)
  3. $x^2 - 8x + c = 0$. In the given equation, $c$ is a constant. If the solutions to the equation are $x = 4 + \sqrt{5}$ and $x = 4 - \sqrt{5}$, what is the value of $c$? (Grid-in)
  4. $y = x^2 - 4x + 3$
    $y = 2x - 6$
    The graphs of the given equations in the system intersect at exactly one point, $(x, y)$, in the $xy$-plane. What is the value of $x$? (Grid-in)
  5. $x^2 + 6x = 40$. The given equation can be rewritten in the form $(x + 3)^2 = p$, where $p$ is a constant. What is the value of $p$? (Grid-in)

Level 4: Very Hard (Slightly beyond SAT)

  1. $2x^2 - bx + 18 = 0$. In the given equation, $b$ is a positive integer constant. If the equation has no real solutions, what is the greatest possible value of $b$? (Grid-in)
  2. The graph of $y = -2x^2 + 24x - c$ in the $xy$-plane has its vertex on the $x$-axis. What is the value of the constant $c$? (Grid-in)
  3. The quadratic function $g$ is defined by $g(x) = a(x - h)^2 + k$, where $a, h,$ and $k$ are constants. The vertex of the graph of $y = g(x)$ in the $xy$-plane is $(4, 10)$. If the graph passes through the point $(2, 2)$, what is the value of $g(6)$? (Grid-in)
  4. $3x^2 - 5x - 7 = 0$. Let $m$ and $n$ be the two solutions to the given equation. What is the value of $m + n$?
    • A) $-\frac{7}{3}$
    • B) $-\frac{5}{3}$
    • C) $\frac{5}{3}$
    • D) $\frac{7}{3}$
  5. The graph of $y = (x - 2)(x + 8)$ is a parabola in the $xy$-plane. A horizontal line defined by the equation $y = c$ intersects the parabola at exactly one point. What is the value of $c$? (Grid-in)

Answer key

The answer to every question, with a brief explanation. Try the sheet first.

Notes for equivalent expressions: Equivalent expressions.

Equivalent expressions and polynomials

Equivalent expressions · 20 questions

Worked Examples: SAT Strategy

Example 1: Easy (Matching Coefficients)

Question: If $3x^2 + kx - 8 = (3x + 4)(x - 2)$ for all values of $x$, what is the value of $k$?
SAT Strategy: The phrase "for all values of $x$" means the left side and right side are mathematically identical. Just expand the right side and match the coefficients!
  1. Expand the right side using FOIL: $3x(x) + 3x(-2) + 4(x) + 4(-2)$.
  2. Simplify: $3x^2 - 6x + 4x - 8$.
  3. Combine like terms: $3x^2 - 2x - 8$.
  4. Match the $x$-coefficient to the left side: $kx = -2x$, so $k = -2$.
  5. Answer: $-2$.

Example 2: Medium (Rational Exponents)

Question: Which of the following expressions is equivalent to $\frac{\sqrt{x^5}}{\sqrt{x}}$ for $x > 0$?
SAT Strategy: Convert all radicals to rational exponents ($x^{a/b}$), then apply standard exponent subtraction rules ($\frac{x^c}{x^d} = x^{c-d}$).
  1. Rewrite the numerator: $\sqrt{x^5} = x^{5/3}$.
  2. Rewrite the denominator: $\sqrt{x} = x^{1/2}$.
  3. Set up the division: $\frac{x^{5/3}}{x^{1/2}}$.
  4. Subtract the exponents: $\frac{5}{3} - \frac{1}{2}$. Find a common denominator (6).
  5. $\frac{10}{6} - \frac{3}{6} = \frac{7}{6}$.
  6. Answer: $x^{7/6}$ (or $\sqrt{x^7}$).

Example 3: Hard (The Remainder Theorem Shortcut)

Question: The expression $\frac{5x + 14}{x + 2}$ can be rewritten in the form $A + \frac{B}{x + 2}$, where $A$ and $B$ are constants. What is the value of $B$?
SAT Strategy: Most students use polynomial long division here. As an AA SL student, you know the Remainder Theorem! The constant $B$ over the divisor is just the remainder. To find the remainder when dividing by $(x - a)$, simply evaluate the polynomial at $x = a$.
  1. The divisor is $(x + 2)$, which means our root to test is $x = -2$.
  2. The numerator is our polynomial $P(x) = 5x + 14$.
  3. Evaluate $P(-2)$ to find the remainder $B$: $5(-2) + 14$.
  4. $-10 + 14 = 4$.
  5. Therefore, the remainder $B = 4$. (And $A$ is obviously 5, the quotient).
  6. Answer: $4$.

Practice Questions

Level 1: Easy (Foundational SAT)

  1. Which expression is equivalent to $(2x^2 - 4x + 3) - (x^2 - 5x - 2)$?
    • A) $x^2 - 9x + 1$
    • B) $x^2 + x + 5$
    • C) $3x^2 - 9x + 1$
    • D) $x^2 - x + 5$
  2. Which expression is equivalent to $4x^3 \cdot 2x^5$?
    • A) $6x^8$
    • B) $8x^{15}$
    • C) $8x^8$
    • D) $6x^{15}$
  3. If $(ax + 2)(bx - 3) = 15x^2 + cx - 6$ for all values of $x$, and $a + b = 8$, what are the two possible values for $c$?
    • A) $1$ and $-1$
    • B) $-1$ and $-19$
    • C) $3$ and $5$
    • D) $1$ and $-9$
  4. Which expression is equivalent to $(x^2y^3)^4$?
    • A) $x^6y^7$
    • B) $x^8y^{12}$
    • C) $x^{16}y^{81}$
    • D) $x^2y^{12}$
  5. What is the value of $(3x - 2)^2$ when expanded?
    • A) $9x^2 - 4$
    • B) $9x^2 + 4$
    • C) $9x^2 - 12x + 4$
    • D) $9x^2 - 6x + 4$

Level 2: Medium (Standard SAT)

  1. Which expression is equivalent to $\frac{x^2 - 16}{x^2 + 8x + 16}$ for $x \neq -4$?
    • A) $-1$
    • B) $\frac{x - 4}{x + 4}$
    • C) $\frac{x + 4}{x - 4}$
    • D) $\frac{1}{8x}$
  2. If $3^x = 10$, what is the value of $3^{x+2}$? (Grid-in)
  3. Which of the following is equivalent to $\sqrt{x^3}$?
    • A) $x^{3/4}$
    • B) $x^{4/3}$
    • C) $x^7$
    • D) $x^{-3/4}$
  4. If $\frac{x^{a^2}}{x^{b^2}} = x^{24}$, $x > 1$, and $a + b = 4$, what is the value of $a - b$? (Grid-in)
  5. The expression $\frac{6x + 11}{2x + 1}$ can be rewritten as $3 + \frac{k}{2x + 1}$, where $k$ is a constant. What is the value of $k$? (Grid-in)

Level 3: Hard (Advanced SAT)

  1. Which of the following is equivalent to $\frac{1}{\frac{1}{x} + \frac{1}{y}}$?
    • A) $x + y$
    • B) $\frac{x + y}{xy}$
    • C) $\frac{xy}{x + y}$
    • D) $xy$
  2. A polynomial $P(x)$ is divided by $(x - 3)$. The quotient is $x^2 - 4x + 2$ and the remainder is $7$. What is $P(3)$? (Grid-in)
  3. Which of the following is a factor of $x^3 - 2x^2 - 9x + 18$?
    • A) $x - 1$
    • B) $x + 2$
    • C) $x + 3$
    • D) $x - 4$
  4. If $8^x \cdot 2^y = 64$ and $3x + y = k$, what is the value of $k$? (Grid-in)
  5. The expression $(x^2 - y^2)^2 - (x^2 + y^2)^2$ is equivalent to which of the following?
    • A) $0$
    • B) $-4x^2y^2$
    • C) $4x^2y^2$
    • D) $-2y^4$

Level 4: Very Hard (Slightly beyond SAT)

  1. The equation $\frac{24x^2 + 25x - 47}{ax - 2} = -8x - 3 - \frac{53}{ax - 2}$ is true for all values of $x \neq \frac{2}{a}$, where $a$ is a constant. What is the value of $a$? (Grid-in)
  2. If $x > 0$ and $x^{1/2} \cdot x^{1/3} \cdot x^{1/6} = x^n$, what is the value of $n$? (Grid-in)
  3. A rational expression simplifies as follows: $\frac{cx^2 + 10x - 8}{2x + 4} = 3x - 2$ for $x \neq -2$. What is the value of $c$? (Grid-in)
  4. Let $p(x) = x^3 - ax^2 + bx - 6$. If $(x - 1)$ and $(x + 2)$ are both factors of $p(x)$, what is the value of $a + b$? (Grid-in)
  5. The expression $\frac{1}{x-1} + \frac{2}{x+2}$ is equivalent to $\frac{cx + d}{x^2 + x - 2}$. What is the value of $c \times d$? (Grid-in)

Answer key

The answer to every question, with a brief explanation. Try the sheet first.

Equivalent expressions: polynomials and rational expressions

Equivalent expressions · 20 questions

Worked Examples: SAT Strategy

Example 1: Easy (Adding/Subtracting Polynomials)

Question: Which expression is equivalent to $(2x^2 - 4x + 1) - (x^2 - 5x - 3)$?
SAT Strategy: The most common mistake is forgetting to distribute the negative sign to every term in the second polynomial.
  1. Distribute the negative: $2x^2 - 4x + 1 - x^2 + 5x + 3$.
  2. Group like terms together: $(2x^2 - x^2) + (-4x + 5x) + (1 + 3)$.
  3. Simplify: $x^2 + x + 4$.
  4. Answer: $x^2 + x + 4$.

Example 2: Medium (Factoring Rational Expressions)

Question: Which expression is equivalent to $\frac{x^2 - 16}{x - 4}$ for $x \neq 4$?
SAT Strategy: Do not attempt to plug in numbers or use a calculator. Always look for the "Difference of Squares" pattern ($a^2 - b^2 = (a-b)(a+b)$) or standard quadratic factoring to cancel out the denominator.
  1. Factor the numerator (Difference of Squares): $x^2 - 16 = (x - 4)(x + 4)$.
  2. Rewrite the fraction: $\frac{(x - 4)(x + 4)}{x - 4}$.
  3. Cancel the common factor of $(x - 4)$ from the top and bottom.
  4. Answer: $x + 4$.

Example 3: Hard (Polynomial Division Concept)

Question: The expression $\frac{3x + 14}{x + 4}$ can be rewritten as $3 + \frac{R}{x + 4}$. What is the value of the constant $R$?
SAT Strategy: You want to manipulate the numerator so that it contains a perfect multiple of the denominator.
  1. The denominator is $x + 4$. To factor out a whole number $3$, we need $3(x + 4) = 3x + 12$ in the numerator.
  2. Rewrite the original numerator $3x + 14$ by splitting the constant: $(3x + 12) + 2$.
  3. Split the fraction into two parts: $\frac{3x + 12}{x + 4} + \frac{2}{x + 4}$.
  4. Simplify the first fraction: $3 + \frac{2}{x + 4}$.
  5. Compare this to the requested format $3 + \frac{R}{x+4}$. It is clear that $R = 2$.
  6. Answer: $2$.

Practice Questions

Level 1: Easy (Foundational SAT)

  1. Which expression is equivalent to $(3x^2 - 5x + 2) + (x^2 + 4x - 6)$?
    • A) $4x^2 - x - 4$
    • B) $4x^2 - 9x - 4$
    • C) $3x^4 - x^2 - 12$
    • D) $2x^2 - x + 8$
  2. Which expression is equivalent to $4x(2x^2 - 3x + 1)$?
    • A) $8x^3 - 12x^2 + 4x$
    • B) $6x^3 - 7x^2 + 5x$
    • C) $8x^2 - 12x + 4$
    • D) $8x^3 - 3x + 1$
  3. Which expression is equivalent to $(x + 6)(x - 6)$?
    • A) $x^2 - 12x - 36$
    • B) $x^2 + 12x - 36$
    • C) $x^2 - 36$
    • D) $x^2 + 36$
  4. Which expression is equivalent to $(5x^3 - 2x) - (3x^3 - 2x)$?
    • A) $2x^3 - 4x$
    • B) $2x^3$
    • C) $8x^3 - 4x$
    • D) $8x^3$
  5. The expression $2(x+3)^2$ is equivalent to which of the following?
    • A) $2x^2 + 18$
    • B) $2x^2 + 9$
    • C) $2x^2 + 6x + 18$
    • D) $2x^2 + 12x + 18$

Level 2: Medium (Standard SAT)

  1. Which expression is equivalent to $\frac{12x^3 - 8x^2 + 4x}{4x}$, where $x \neq 0$?
    • A) $3x^2 - 2x$
    • B) $3x^2 - 2x + 1$
    • C) $8x^2 - 4x + 1$
    • D) $3x^3 - 2x^2 + x$
  2. Which expression is equivalent to $(2x - 5)(3x + 4)$?
    • A) $6x^2 - 20$
    • B) $6x^2 - 7x - 20$
    • C) $6x^2 + 7x - 20$
    • D) $6x^2 - 23x - 20$
  3. Which expression is equivalent to $\frac{x^2 - 25}{x - 5}$ for $x \neq 5$?
    • A) $x - 5$
    • B) $x + 5$
    • C) $x - 20$
    • D) $x^2 - 5$
  4. Which expression is equivalent to $x^2y - 3xy^2 + 2x^2y$?
    • A) $3x^2y - 3xy^2$
    • B) $-xy^2$
    • C) $3x^4y^2 - 3xy^2$
    • D) $0$
  5. The expression $3x^2 + 21x + 30$ can be rewritten as $3(x+a)(x+b)$, where $a$ and $b$ are positive constants. What is the value of $a \times b$? (Grid-in)

Level 3: Hard (Advanced SAT)

  1. Which expression is equivalent to $\frac{42a}{k} + 42ak$, where $k > 0$?
    • A) $\frac{84a}{k}$
    • B) $\frac{84ak^2}{k}$
    • C) $\frac{42a(k+1)}{k}$
    • D) $\frac{42a(k^2+1)}{k}$
  2. If $ax^2 + bx + c = (2x - 3)(5x + 1)$ for all values of $x$, what is the value of $b$? (Grid-in)
  3. Which expression is equivalent to $\frac{2x^2 + 7x + 5}{x+1}$ for $x \neq -1$?
    • A) $2x + 5$
    • B) $2x + 7$
    • C) $2x - 5$
    • D) $x + 5$
  4. The expression $\frac{x}{y} + \frac{y}{x}$ is equivalent to which of the following?
    • A) $1$
    • B) $\frac{x+y}{xy}$
    • C) $\frac{x^2+y^2}{xy}$
    • D) $\frac{(x+y)^2}{xy}$
  5. If $(x - 4)^2 - 16 = x^2 - px$ for all values of $x$, what is the value of $p$? (Grid-in)

Level 4: Very Hard (Slightly beyond SAT)

  1. The expression $2x^2 + bx - 45$, where $b$ is a constant, can be rewritten as $(hx+k)(x+j)$, where $h$, $k$, and $j$ are integer constants. Which of the following must be an integer?
    • A) $\frac{b}{h}$
    • B) $\frac{b}{k}$
    • C) $\frac{45}{h}$
    • D) $\frac{45}{k}$
  2. The expression $\frac{5x + 13}{x+2}$ can be rewritten in the form $A + \frac{B}{x+2}$, where $A$ and $B$ are constants. What is the value of $B$? (Grid-in)
  3. If $cx^2 + 10x + d = (ax+b)^2$ for all values of $x$, where $a,b,c,d$ are positive constants, what is the value of $c \times d$? (Grid-in)
  4. Which expression is equivalent to $(x^2 + y^2)^2 - (x^2 - y^2)^2$?
    • A) $0$
    • B) $2y^4$
    • C) $4x^2y^2$
    • D) $2x^4 + 2y^4$
  5. The expression $\frac{1}{x+3} + \frac{1}{x-3}$ is equivalent to $\frac{ax}{x^2 - 9}$ for $x \neq 3$ and $x \neq -3$. What is the value of $a$? (Grid-in)

Answer key

The answer to every question, with a brief explanation. Try the sheet first.

Notes for exponential functions: Nonlinear functions.

Exponential functions: set 1

Exponential functions · 20 questions

Worked Examples: SAT Strategy

Example 1: Easy (Translating Percentages into Bases)

Question: At the time an article was first featured on a news website, there were 40 comments. At the end of each hour, the number of comments had increased by $190%$ of the number of comments at the end of the previous hour. Which equation best represents the estimated number of comments $C$ after $t$ hours?
SAT Strategy: A common trap is using the percentage itself as the base. The base of an exponential function is always $1 + r$ (for growth) or $1 - r$ (for decay).
  1. Identify the initial value ($a$): $40$.
  2. Identify the rate of increase ($r$): $190%$ translates to $1.90$.
  3. Calculate the base ($1 + r$): $1 + 1.90 = 2.90$.
  4. Set up the model $C = a(b)^t$: $C = 40(2.9)^t$.
  5. Answer: $C = 40(2.9)^t$.

Example 2: Medium (Extracting the Decay Rate)

Question: The function $f(x) = 5470(0.64)^{x/12}$ gives the value, in dollars, of a piece of equipment after $x$ months of use. If the value of the equipment decreases each year by $p%$ of its value the preceding year, what is the value of $p$?
SAT Strategy: The exponent $x/12$ means the full decay factor of $0.64$ occurs exactly once every $12$ months (which is $1$ year). You simply need to translate the base $0.64$ back into a percentage decrease.
  1. Because the base is less than $1$, it is exponential decay.
  2. The base $b = 1 - r$. Therefore, $0.64 = 1 - r$.
  3. Solve for $r$: $r = 1 - 0.64 = 0.36$.
  4. Convert the decimal rate back to a percentage: $0.36 \implies 36%$.
  5. Answer: $36$.

Example 3: Hard (Constructing Fractional Exponents)

Question: A population of bacteria initially contains $3,000$ cells. The population doubles every $15$ minutes. Which of the following functions models the population, $P(t)$, after $t$ hours?
SAT Strategy: Pay close attention to the units of time! The growth happens in minutes, but the variable $t$ is in hours. You must convert the units so they match.
  1. The initial value $a = 3000$. The base $b = 2$ (since it "doubles").
  2. The doubling happens every 15 minutes. In terms of hours, 15 minutes is $\frac{15}{60} = 0.25$ hours.
  3. The exponent must divide the total time $t$ by the time it takes for one cycle ($k$).
  4. Exponent $= \frac{t}{k} = \frac{t}{0.25} = 4t$. (This makes sense: it doubles 4 times per hour).
  5. Answer: $P(t) = 3000(2)^{4t}$.

Practice Questions

Level 1: Easy (Foundational SAT)

  1. A savings account starts with an initial deposit of $\$500$. The account earns $3%$ interest, compounded annually. Which function $A(t)$ represents the amount of money in the account after $t$ years?
    • A) $A(t) = 500(0.03)^t$
    • B) $A(t) = 500(1.03)^t$
    • C) $A(t) = 500(1.30)^t$
    • D) $A(t) = 500 + 1.03t$
  2. The population of a small town decreases by $2%$ every year. If the current population is $15,000$, which expression models the population after $t$ years?
    • A) $15000(0.98)^t$
    • B) $15000(0.02)^t$
    • C) $15000(1.02)^t$
    • D) $15000 - 0.02t$
  3. A painting was purchased for $\$1,200$. Its value $V(t)$, in dollars, $t$ years after the purchase is modeled by $V(t) = 1200(1.15)^t$. By what percentage does the value of the painting increase each year? (Grid-in)
  4. The function $P(x) = 250(0.88)^x$ models the number of pests in a garden $x$ weeks after a treatment is applied. What is the best interpretation of the number $0.88$ in this context?
    • A) The number of pests decreases by $88%$ each week.
    • B) The number of pests remaining is $88%$ of the amount from the previous week.
    • C) The number of pests decreases by $0.88$ each week.
    • D) The treatment was $88%$ effective overall.
  5. A rare coin is currently worth $\$400$. If its value triples every decade, which equation represents the coin's value $V(d)$ after $d$ decades?
    • A) $V(d) = 400(3)^d$
    • B) $V(d) = 400(\frac{1}{3})^d$
    • C) $V(d) = 3(400)^d$
    • D) $V(d) = 400 + 3^d$

Level 2: Medium (Standard SAT)

  1. The value of a stock portfolio $V$, in dollars, is modeled by the equation $V(t) = 5000(1.045)^{t/12}$, where $t$ is the number of months since the portfolio was opened. What is the percentage increase in the portfolio's value exactly one year after it was opened? (Grid-in)
  2. A radioactive isotope has a half-life of $8$ days. If a scientist begins with $200$ grams of the isotope, which function $M(d)$ models the mass of the isotope remaining after $d$ days?
    • A) $M(d) = 200(0.5)^{8d}$
    • B) $M(d) = 200(0.5)^{d/8}$
    • C) $M(d) = 200(8)^{d/0.5}$
    • D) $M(d) = 8(0.5)^{d/200}$
  3. A car depreciates in value by $15%$ every year. If the original cost of the car was $\$24,000$, which of the following represents the car's value $V(t)$ after $t$ months?
    • A) $V(t) = 24000(0.85)^{12t}$
    • B) $V(t) = 24000(0.85)^{t/12}$
    • C) $V(t) = 24000(1.15)^{t/12}$
    • D) $V(t) = 24000(0.15)^{t/12}$
  4. The function $h(x) = 150(2.4)^{x}$ models the number of users on an app $x$ weeks after its launch. The number of users increases by $p%$ each week. What is the value of $p$? (Grid-in)
  5. The expression $12(1.08)^t$ models the height, in centimeters, of a sapling $t$ weeks after it is planted. If the expression is rewritten in the form $a(b)^{t/7}$ to model the height after $d$ days, what is the value of $b$?
    • A) $\frac{1.08}{7}$
    • B) $1.08$
    • C) $(1.08)^7$
    • D) $\sqrt{1.08}$

Level 3: Hard (Advanced SAT)

  1. The population of a city is given by the function $P(t) = 45000(1.21)^{t/10}$, where $t$ is the number of years since 2000. Which of the following is the best interpretation of the number $1.21$ in this context?
    • A) The population increases by $21%$ every year.
    • B) The population increases by $121%$ every $10$ years.
    • C) The population increases by $21%$ every $10$ years.
    • D) The population multiplies by $1.21$ every year.
  2. A colony of insects has an initial population of $N_0$. The population is observed to triple every $k$ hours. Which of the following equations gives the population $P(h)$ after $h$ hours?
    • A) $P(h) = N_0(3)^{h/k}$
    • B) $P(h) = N_0(3)^{k/h}$
    • C) $P(h) = N_0(\frac{1}{3})^{h/k}$
    • D) $P(h) = 3N_0(k)^{h}$
  3. A biologist models the growth of a fungus using $A(t) = 50(2)^{3t}$, where $t$ is the time in days. The biologist wants to rewrite the function in the form $A(t) = a(b)^t$. What is the value of $b$? (Grid-in)
  4. The function $f(x) = 100(0.81)^x$ can be rewritten in the form $f(x) = 100(b)^{2x}$, where $b$ is a constant. What is the value of $b$? (Grid-in)
  5. A company's revenue $R$, in millions of dollars, $t$ years after $2010$ is modeled by $R(t) = 14(1.06)^t$. The company decides to create a new model $S(m)$ where $m$ is the number of months after $2010$. Which equation is the correct model for $S(m)$?
    • A) $S(m) = 14(1.06)^{m/12}$
    • B) $S(m) = 14(1.06)^{12m}$
    • C) $S(m) = 14(\frac{1.06}{12})^m$
    • D) $S(m) = \frac{14}{12}(1.06)^{m/12}$

Level 4: Very Hard (Slightly beyond SAT)

  1. A scientist models the decay of a substance with the function $D(t) = 800(0.5)^{t/6}$, where $t$ is the time in hours. The scientist wants to rewrite the model to show the percentage of the substance that decays each hour. Which of the following expressions represents the percentage that decays per hour?
    • A) $100(1 - (0.5)^{1/6})$
    • B) $100(0.5)^{1/6}$
    • C) $100(1 - (0.5)^6)$
    • D) $\frac{50}{6}$
  2. In the function $P(x) = a(b)^x$, $P(2) = 144$ and $P(4) = 324$. Assuming $b > 0$, what is the value of the initial amount $a$? (Grid-in)
  3. The population of Town A is modeled by $P_A(t) = 2000(1.4)^t$ and the population of Town B is modeled by $P_B(t) = 3000(1.2)^t$, where $t$ is measured in years. In the model for the ratio of the population of Town A to Town B, $R(t) = \frac{P_A(t)}{P_B(t)}$, the ratio grows exponentially by $q%$ per year. What is the value of $q$ to the nearest tenth? (Grid-in)
  4. A bank account earns a certain interest rate compounded annually. After $5$ years, the account balance has grown by exactly $30%$ of its original value. Which expression represents the annual interest rate as a decimal?
    • A) $0.30/5$
    • B) $1.30^{1/5}$
    • C) $1.30^{1/5} - 1$
    • D) $1 - 0.30^{1/5}$
  5. If $f(x) = 10(2^x)$ and $g(x) = f(x + 3)$, what is the ratio of $g(x)$ to $f(x)$ for any real value of $x$? (Grid-in)

Answer key

The answer to every question, with a brief explanation. Try the sheet first.

Exponential functions: set 2

Exponential functions · 20 questions

Worked Examples: SAT Strategy

Example 1: Easy (Interpreting Constants)

Question: The function $P(t) = 1,800(1.02)^t$ gives the estimated number of marine mammals in a certain area, where $t$ is the number of years since a study began. What is the best interpretation of $P(0) = 1,800$ in this context?
SAT Strategy: Recognize the standard exponential form $y = a(b)^x$, where $a$ is the initial value (when $x=0$) and $b$ is the growth/decay factor.
  1. $P(0)$ represents the value of the function when $t = 0$.
  2. Since $t$ is the number of years since the study began, $t = 0$ means the exact moment the study started.
  3. Answer: The estimated number of marine mammals in the area was 1,800 when the study began.

Example 2: Medium (Translating Percentages to Bases)

Question: At the time an article was first featured on a website, there were 40 comments. At the end of each hour, the number of comments had increased by 190% of the number of comments at the end of the previous hour. Which equation represents this model?
SAT Strategy: A percentage increase is always added to 100% (or $1$).
  1. The initial amount is $a = 40$.
  2. An increase of $190%$ means the new amount is $100% + 190% = 290%$ of the previous amount.
  3. Convert $290%$ to a decimal base $b$ by dividing by 100: $b = 2.90$.
  4. Substitute into $C = a(b)^t$.
  5. Answer: $C = 40(2.9)^t$.

Example 3: Hard (Fractional Exponents and Doubling Time)

Question: The function $f(t) = 60,000(2)^{\frac{t}{410}}$ gives the number of bacteria in a population $t$ minutes after an initial observation. How much time, in minutes, does it take for the number of bacteria in the population to double?
SAT Strategy: When an exponential equation is in the form $y = a(2)^{\frac{t}{k}}$, the value $k$ is the exact time it takes for the population to double.
  1. For the population to double, the multiplier must be exactly $2^1$.
  2. This means the exponent must equal 1: $\frac{t}{410} = 1$.
  3. Multiply both sides by 410: $t = 410$.
  4. Answer: 410.

Practice Questions

Level 1: Easy (Foundational SAT)

  1. The function $f$ is defined by $f(x) = 270(0.1)^x$. What is the value of $f(0)$? (Grid-in)
  2. The population of Greenville increased by 7% from 2015 to 2016. If the 2016 population is $k$ times the 2015 population, what is the value of $k$?
    • A) $0.07$
    • B) $0.93$
    • C) $1.07$
    • D) $7$
  3. The function $h$ is defined by $h(x) = 5(2)^x$. What is the value of $h(3)$? (Grid-in)
  4. A company opens an account with an initial balance of $\$36,100.00$. The account earns 5% interest annually, and no additional deposits or withdrawals are made. Which equation could define $A(t)$, the account balance $t$ years after the account is opened?
    • A) $A(t) = 36,100(1.05)^t$
    • B) $A(t) = 36,100(0.05)^t$
    • C) $A(t) = 36,100(1.5)^t$
    • D) $A(t) = 36,100(0.95)^t$
  5. What is the $y$-intercept of the graph of $y = 12(0.8)^x$ in the $xy$-plane?
    • A) $(0, 0.8)$
    • B) $(0, 9.6)$
    • C) $(0, 12)$
    • D) $(12, 0)$

Level 2: Medium (Standard SAT)

  1. For $x > 0$, the function $f$ is defined as follows: $f(x)$ equals $201%$ of $x$. Which of the following could describe this function?
    • A) Decreasing exponential
    • B) Decreasing linear
    • C) Increasing exponential
    • D) Increasing linear
  2. The table below shows selected values for the function $g$. {|c|c|} $x$ & $g(x)$
    $0$ & $3$
    $1$ & $12$
    $2$ & $48$
    Which of the following equations defines $g(x)$?
    • A) $g(x) = 3(4)^x$
    • B) $g(x) = 4(3)^x$
    • C) $g(x) = 3x + 9$
    • D) $g(x) = 9x + 3$
  3. The function $h$ is defined by $h(x) = 10(3)^x$. In the $xy$-plane, the graph of $y = p(x)$ is the result of shifting the graph of $y = h(x)$ down 2 units. What is the $y$-coordinate of the $y$-intercept of the graph of $p$? (Grid-in)
  4. A radioactive substance has a half-life of 5 days. If the initial amount of the substance is 100 grams, which equation models the amount remaining, $R$, after $d$ days?
    • A) $R = 100(0.5)^{5d}$
    • B) $R = 100(0.5)^{\frac{d}{5}}$
    • C) $R = 100(5)^{\frac{d}{0.5}}$
    • D) $R = 50(0.5)^d$
  5. $2^{3x - 1} = 32$. What is the value of $x$? (Grid-in)

Level 3: Hard (Advanced SAT)

  1. The function $f(x) = 5,470(0.64)^{\frac{x}{12}}$ gives the value, in dollars, of a piece of equipment after $x$ months of use. If the value of the equipment decreases each year by $p%$ of its value the preceding year, what is the value of $p$? (Grid-in)
  2. The functions $f$ and $g$ are defined for $x \ge 0$ by the equations $f(x) = 18(1.25)^x + 41$ and $g(x) = 9(0.73)^x$. Which of the following equations displays, as a constant or coefficient, the maximum value of the function it defines?
    • A) I only
    • B) II only
    • C) I and II
    • D) Neither I nor II
  3. Which of the following expressions is equivalent to $2^{x+3} - 2^x$?
    • A) $2^3$
    • B) $3(2^x)$
    • C) $7(2^x)$
    • D) $8(2^x)$
  4. An exponential function $f$ is given by $f(x) = a(b)^x$, where $a$ and $b$ are positive constants. If $f(0) = 5$ and $f(2) = 45$, what is the value of $a + b$? (Grid-in)
  5. If $3^{x^2 - 4x} = 1$, what is the positive solution for $x$? (Grid-in)

Level 4: Very Hard (Slightly beyond SAT)

  1. A savings account earns $8%$ annual interest, compounded yearly. If the initial deposit is $\$500$, the amount $A$ in the account after $t$ years is $A = 500(1.08)^t$. Which of the following equations is equivalent and shows the approximate monthly interest rate?
    • A) $A = 500(1.0064)^{12t}$
    • B) $A = 500(1.08)^{12t}$
    • C) $A = 500(0.0064)^{12t}$
    • D) $A = 500(1.0064)^{\frac{t}{12}}$
  2. $P(t) = 260(1.04)^{\frac{t}{6}}$. The function $P$ models the population of a city $t$ years after 2003. According to the model, the population is predicted to increase by $4%$ every $n$ months. What is the value of $n$? (Grid-in)
  3. The function $y = 20(3)^{\frac{x}{2}}$ relates the variables $x$ and $y$. If the value of $x$ is increased by 4, by what factor is the value of $y$ multiplied?
    • A) $3$
    • B) $6$
    • C) $9$
    • D) $81$
  4. In the $xy$-plane, the graphs of $y = 3^x$ and $y = 2(3^x) - 9$ intersect at the point $(x, y)$. What is the value of $y$? (Grid-in)
  5. The function $f$ is defined by $f(x) = 2.2^x$. The function $g$ is defined by $g(x) = f(x+3)$. The equation $g(x) = k(2.2^x)$ is true for all $x$, where $k$ is a constant. What is the value of $k$? (Round your answer to the nearest hundredth if necessary). (Grid-in)

Answer key

The answer to every question, with a brief explanation. Try the sheet first.

Notes for nonlinear systems of equations: Nonlinear equations and systems.

Nonlinear systems: set 1

Nonlinear systems of equations · 20 questions

Worked Examples: SAT Strategy

Example 1: Easy (Standard Intersection)

Question: The graphs of $y = x^2 - 5x + 6$ and $y = x - 3$ intersect at two points in the $xy$-plane. What is the $x$-coordinate of the intersection point that lies in Quadrant I?
SAT Strategy: Set the equations equal to each other, move everything to one side to form a new quadratic, and factor.
  1. Set equal: $x^2 - 5x + 6 = x - 3$.
  2. Move to one side: $x^2 - 6x + 9 = 0$.
  3. Factor the perfect square trinomial: $(x - 3)^2 = 0$.
  4. Solve: $x = 3$.
  5. Wait, there is only one intersection point, which means the (Full worked solution in mark scheme.)
  6. Answer: $3$.

Example 2: Medium (Tangent Lines and the Discriminant)

Question: The system of equations $y = 2x^2 - 8x + 11$ and $y = mx + 3$ has exactly one real solution. If $m > 0$, what is the value of $m$?
SAT Strategy: "Exactly one solution" means the line is tangent to the parabola. When you merge the equations, the discriminant ($\Delta = b^2 - 4ac$) of the new quadratic must equal exactly zero.
  1. Set equal: $2x^2 - 8x + 11 = mx + 3$.
  2. Group the $x$ terms together: $2x^2 - (8 + m)x + 8 = 0$.
  3. Apply $\Delta = 0$: $(-(8 + m))^2 - 4(2)(8) = 0$.
  4. Simplify: $(8 + m)^2 - 64 = 0 \implies (8 + m)^2 = 64$.
  5. Take the square root: $8 + m = \pm 8$.
  6. Solve for $m$: $m = 0$ or $m = -16$. (Wait, neither is $>0$. (Full worked solution in mark scheme.)

Example 3: Hard (Sum of Intersections Shortcut)

Question: The line $y = 5x + c$ intersects the parabola $y = -2x^2 + 9x + 14$ at exactly two points. What is the sum of the $x$-coordinates of these two intersection points?
SAT Strategy: Do not try to solve for the roots using the quadratic formula! Merge the equations and use the Sum of Roots shortcut ($x_1 + x_2 = -b/a$). Notice that the constant $c$ doesn't even matter for the sum!
  1. Set equal: $-2x^2 + 9x + 14 = 5x + c$.
  2. Move to one side: $-2x^2 + 4x + (14 - c) = 0$.
  3. Identify $a = -2$ and $b = 4$.
  4. Apply the sum formula: $-\frac{b}{a} = -\frac{4}{-2} = 2$.
  5. Answer: $2$.

Practice Questions

Level 1: Easy (Foundational SAT)

  1. The graphs of $y = x^2$ and $y = 16$ intersect at two points in the $xy$-plane. If $(x, y)$ is an intersection point and $x > 0$, what is the value of $x$? (Grid-in)
  2. The system of equations $y = x^2 - 4x$ and $y = x$ has two solutions, $(x_1, y_1)$ and $(x_2, y_2)$. If $x_1 < x_2$, what is the value of $x_2$?
    • A) $0$
    • B) $3$
    • C) $4$
    • D) $5$
  3. The line $y = 3x - 7$ and the parabola $y = 2x^2 + 3x - 7$ intersect at exactly one point. What is the $x$-coordinate of this point? (Grid-in)
  4. The system of equations $x^2 + y = 20$ and $y = 4$ intersects at two points. What is the positive difference between the $x$-coordinates of these two points?
    • A) $4$
    • B) $8$
    • C) $16$
    • D) $20$
  5. For what positive value of $x$ does the graph of $y = (x - 3)^2$ intersect the line $y = 9$? (Grid-in)

Level 2: Medium (Standard SAT)

  1. The line $y = c$ is tangent to the parabola $y = x^2 - 6x + 8$ in the $xy$-plane. What is the value of the constant $c$?
    • A) $-1$
    • B) $0$
    • C) $3$
    • D) $8$
  2. The graphs of $y = -x^2 + 7x$ and $y = 10$ intersect at two points. What is the sum of the $x$-coordinates of these two points of intersection? (Grid-in)
  3. The parabola $y = x^2 - 2x + 3$ intersects the line $y = kx + 3$ at $x = 0$ and $x = 5$. What is the value of $k$?
    • A) $2$
    • B) $3$
    • C) $5$
    • D) $18$
  4. How many points of intersection exist between the graphs of $y = x^2 + x + 1$ and $y = -x^2 + 3$?
    • A) $0$
    • B) $1$
    • C) $2$
    • D) Infinitely many
  5. The system of equations $y = 2x - 3$ and $y = x^2 - 4x + 6$ intersects at exactly one point, $(x, y)$. What is the value of $x$? (Grid-in)

Level 3: Hard (Advanced SAT)

  1. The system of equations $y = x^2 - kx + 7$ and $y = 3x - 2$ has exactly one real solution. If $k > 0$, what is the value of $k$? (Grid-in)
  2. The circle $x^2 + y^2 = 13$ and the line $y = x + 1$ intersect at two points in the $xy$-plane. For the point of intersection in the first quadrant $(x > 0, y > 0)$, what is the value of $x + y$?
    • A) $2$
    • B) $3$
    • C) $5$
    • D) $13$
  3. How many real solutions exist for the system of equations $y = 2x^2 + 5$ and $y = -2x^2 - 5$?
    • A) $0$
    • B) $1$
    • C) $2$
    • D) $4$
  4. The graphs of $y = x^2 - 8x + 15$ and $x - y = 3$ intersect at two points. What is the value of the larger $x$-coordinate of the two intersection points? (Grid-in)
  5. The system $y = 2x^2 - 4x + 11$ and $y = mx + 3$ intersects at exactly one point. If $m > 0$, what is the value of $m$? (Grid-in)

Level 4: Very Hard (Slightly beyond SAT)

  1. The system of equations $y = 3x^2 + 6x + k$ and $y = 2x + 1$ has no real solutions. Which of the following inequalities correctly defines all possible values of the constant $k$?
    • A) $k < \frac{7}{3}$
    • B) $k > \frac{7}{3}$
    • C) $k < -\frac{1}{3}$
    • D) $k > -\frac{1}{3}$
  2. The line $y = 3x$ intersects the circle $x^2 + y^2 = 40$ at two points. What is the straight-line distance between these two points of intersection?
    • A) $4\sqrt{10}$
    • B) $8\sqrt{5}$
    • C) $20$
    • D) $40$
  3. The horizontal line $y = a$ intersects the parabola $y = (x - 2)^2 - 5$ at two points, $P$ and $Q$. If the distance between point $P$ and point $Q$ is exactly $8$ units, what is the value of $a$? (Grid-in)
  4. The line $y = cx$ intersects the parabola $y = -x^2 + 6x + 10$ at two points in the $xy$-plane. If the sum of the $x$-coordinates of the two intersection points is exactly $0$, what is the value of $c$? (Grid-in)
  5. The parabola $y = x^2 - k$ and the parabola $y = -x^2 + k$ intersect at exactly two points in the $xy$-plane. If $k > 0$, which of the following represents the positive difference between the $x$-coordinates of the intersection points?
    • A) $k$
    • B) $2\sqrt{k}$
    • C) $\sqrt{2k}$
    • D) $2k$

Answer key

The answer to every question, with a brief explanation. Try the sheet first.

Nonlinear systems: set 2

Nonlinear systems of equations · 20 questions

Worked Examples: SAT Strategy

Example 1: Easy (Basic Substitution)

Question: The graphs of the equations $y = x^2 - 5x + 10$ and $y = 4$ intersect at two points in the $xy$-plane. What is the $x$-coordinate of the rightmost point of intersection?
SAT Strategy: Substitute the known $y$-value into the quadratic equation and solve for $x$.
  1. Substitute $y = 4$: $4 = x^2 - 5x + 10$.
  2. Set the equation to zero: $0 = x^2 - 5x + 6$.
  3. Factor the quadratic: $0 = (x - 2)(x - 3)$.
  4. The two points of intersection are at $x = 2$ and $x = 3$.
  5. The rightmost point has the larger $x$-value.
  6. Answer: $3$.

Example 2: Medium (Setting Equations Equal)

Question: The system of equations $y = x^2 + 3x - 5$ and $y = x + 3$ intersects at exactly two points, $(x_1, y_1)$ and $(x_2, y_2)$. What is the value of $x_1 + x_2$?
SAT Strategy: Since both equations are equal to $y$, set them equal to each other to create a single quadratic equation.
  1. Set them equal: $x^2 + 3x - 5 = x + 3$.
  2. Move all terms to the left side: $x^2 + 2x - 8 = 0$.
  3. The question asks for the sum of the $x$-coordinates ($x_1 + x_2$). You can factor to find the roots ($x = -4$ and $x = 2$, sum is $-2$) OR use the "sum of roots" shortcut: $-b/a$.
  4. Sum of roots $= -\frac{2}{1} = -2$.
  5. Answer: $-2$.

Example 3: Hard (Tangent Lines and Discriminants)

Question: The graph of $y = x^2 - 6x + 14$ and the line $y = mx - 2$ intersect at exactly one point. If $m > 0$, what is the value of $m$?
SAT Strategy: "Exactly one point of intersection" means the line is tangent to the parabola. If you set them equal, the resulting quadratic MUST have a discriminant ($\Delta = b^2 - 4ac$) of $0$.
  1. Set them equal: $x^2 - 6x + 14 = mx - 2$.
  2. Move all terms to the left side: $x^2 - 6x - mx + 16 = 0$.
  3. Group the $x$ terms to find $b$: $x^2 - (6 + m)x + 16 = 0$.
  4. Identify $a = 1$, $b = -(6 + m)$, $c = 16$.
  5. Set discriminant to zero: $(-(6+m))^2 - 4(1)(16) = 0 \implies (6+m)^2 - 64 = 0$.
  6. Solve for $m$: $(6+m)^2 = 64 \implies 6+m = \pm 8$.
  7. Therefore, $m = 2$ or $m = -14$. Since $m > 0$, $m = 2$.
  8. Answer: $2$.

Practice Questions

Level 1: Easy (Foundational SAT)

  1. The graphs of $y = x^2 - 1$ and $y = 8$ intersect at two points. What is the positive $x$-coordinate of their intersection? (Grid-in)
  2. The system of equations $x = 5$ and $y = x^2 - 2x + 1$ intersects at exactly one point $(x, y)$. What is the value of $y$? (Grid-in)
  3. How many real solutions does the system of equations $y = x^2 + 5$ and $y = -2$ have?
    • A) Zero
    • B) Exactly one
    • C) Exactly two
    • D) Infinitely many
  4. The graphs of $y = x^2 + 2x$ and $y = -x^2 + 4$ are shown in the $xy$-plane. Which of the following best describes the maximum number of times these two parabolas can intersect?
    • A) 1
    • B) 2
    • C) 3
    • D) 4
  5. The line $y - x = 4$ intersects the parabola $y = x^2 + 4$ at exactly two points. One of those points is $(0, 4)$. What is the $x$-coordinate of the other point of intersection? (Grid-in)

Level 2: Medium (Standard SAT)

  1. The graphs of $y = x^2 - 4$ and $y = 2x - 4$ intersect at two points. What is the sum of the $x$-coordinates of the intersection points?
    • A) $0$
    • B) $2$
    • C) $4$
    • D) $8$
  2. The system of equations $y = x^2 + 5x + 6$ and $y = 2x + 10$ intersects at two points. What is the $x$-coordinate of the intersection point that has a negative $x$-value? (Grid-in)
  3. A circle in the $xy$-plane has the equation $x^2 + y^2 = 16$. A horizontal line has the equation $y = 4$. The line intersects the circle at exactly one point $(x, y)$. What is the value of $x$? (Grid-in)
  4. The vertex of the parabola $y = -(x - 3)^2 + 5$ is touched by the line $y = c$ at exactly one point. What is the value of $c$?
    • A) $-3$
    • B) $3$
    • C) $5$
    • D) $9$
  5. The graphs of $y = 2x^2 - 4x + 5$ and $y = 2x^2 + 3x - 9$ intersect at exactly one point. What is the $x$-coordinate of this intersection? (Grid-in)

Level 3: Hard (Advanced SAT)

  1. $y = x^2 - 8x + 20$
    $y = c$
    In the system of equations above, $c$ is a constant. If the system has exactly one solution, what is the value of $c$? (Grid-in)
  2. $y = x^2 + 4x + 9$
    $y = mx$
    In the system of equations above, $m$ is a constant. If the system has exactly one solution and $m > 0$, what is the value of $m$? (Grid-in)
  3. $y = -x^2 + 5x$
    $y = 5$
    How many distinct real solutions does the given system of equations have?
    • A) Zero
    • B) Exactly one
    • C) Exactly two
    • D) Infinitely many
  4. A circle in the $xy$-plane is defined by the equation $(x - 2)^2 + (y - 3)^2 = 25$. The line $y = 3$ intersects the circle at two points, $A$ and $B$. What is the length of line segment $AB$?
    • A) $5$
    • B) $10$
    • C) $20$
    • D) $25$
  5. $y = x^2 + cx + 10$
    $y = 2x + 1$
    In the system of equations above, $c$ is a constant. The graphs of the equations intersect at exactly one point in the $xy$-plane. Which of the following could be the value of $c$?
    • A) $-6$
    • B) $2$
    • C) $8$
    • D) $10$

Level 4: Very Hard (Slightly beyond SAT)

  1. The parabola $y = (x - 3)^2 - 2$ intersects the line $y = 4x + c$ at exactly one point. What is the value of the constant $c$? (Grid-in)
  2. $x^2 + y^2 = 25$
    $y = x + 1$
    The graphs of the given system of equations intersect at two points in the $xy$-plane. What is the sum of the $x$-coordinates of the two intersection points? (Grid-in)
  3. $y = 3x^2 - 7x + 4$
    $y = 2x^2 - 2x - 2$
    The graphs of the given equations intersect at two points in the $xy$-plane, $(x_1, y_1)$ and $(x_2, y_2)$. What is the value of $x_1 \times x_2$? (Grid-in)
  4. $x = y^2 - 4y + 4$
    $x = 4$
    The graphs of the given equations intersect at two points. What is the sum of the $y$-coordinates of the intersection points?
    • A) $0$
    • B) $2$
    • C) $4$
    • D) $8$
  5. The graph of a parabola in the $xy$-plane has its vertex at $(2, 8)$ and its $y$-intercept at $(0, 4)$. A line $L$ intersects the parabola exactly at its vertex and its $y$-intercept. What is the $x$-intercept of line $L$? (Grid-in)

Answer key

The answer to every question, with a brief explanation. Try the sheet first.

Exponentials, polynomials and complex roots

Stretch sheets · 10 questions · 10 SAT-style advanced-math questions that flex the HL AA syllabus: exponential decay, polynomial factoring, and quadratics with complex roots.

Practice Questions

Level 1: Easy (Foundational)

  1. Determine $2^{-3}$ as an exact fraction.
    A) $\tfrac{1}{6}$
    B) $\tfrac{1}{8}$
    C) $-8$
    D) $-6$
  2. Factor completely: $x^2 - 9$.
    A) $(x-3)(x+3)$
    B) $(x-3)^2$
    C) $(x+3)^2$
    D) $(x-9)(x+1)$
  3. Determine the value of $\log_{10}(1000)$. (SPR)

Level 2: Medium (Standard)

  1. If $p(x) = x^3 - 7x + 6$ and $p(1) = 0$, which factor of $p$ must exist?
    A) $x-1$
    B) $x+1$
    C) $x-6$
    D) $x+6$
  2. Solve $x^2 + 2x + 5 = 0$ over $\mathbb{C}$. Which pair are the roots?
    A) $-1 \pm 2i$
    B) $1 \pm 2i$
    C) $-1 \pm i$
    D) $2 \pm i$
  3. A population $P$ satisfies $P(t) = 500 \cdot 2^{t/3}$ where $t$ is in years. Determine the exact population when $t=6$. (SPR)
  4. The graph of $y=2^{x}$ is transformed to $y = 2^{x-3}$. Which best describes the change?
    A) Shift left 3
    B) Shift right 3
    C) Shift up 3
    D) Shift down 3

Level 3: Hard (Advanced)

  1. $p(x) = x^3 + ax^2 - x - 3$ has a factor $(x-1)$. Determine the exact value of $a$.
    A) $1$
    B) $2$
    C) $3$
    D) $-1$
  2. If $\log_2 x + \log_2 (x-2) = 3$, determine the exact positive value of $x$.
    A) $1+\sqrt{5}$
    B) $1+\sqrt{7}$
    C) $1+\sqrt{9}$
    D) $4$
  3. Determine the exact modulus $|z|$ of the complex number $z = 3 - 4i$. (SPR)

Answer key

The answer to every question, with a brief explanation. Try the sheet first.

Exponentials, logarithms and modelling

Stretch sheets · 10 questions · 10 SAT-style questions on exponential/logarithmic modelling — HL AI territory (population growth, radioactive decay, financial models).

Practice Questions

Level 1: Easy (Foundational)

  1. Determine $\log_{10}(100)$.
    A) $1$
    B) $2$
    C) $10$
    D) $100$
  2. What is $3^4$?
    A) $12$
    B) $27$
    C) $64$
    D) $81$
  3. A population of $200$ doubles every $5$ years. Determine the exact population after $10$ years. (SPR)

Level 2: Medium (Standard)

  1. A radioactive sample decays with half-life $4$ years. Starting at $80$ g, determine the exact mass after $12$ years.
    A) $5$ g
    B) $10$ g
    C) $20$ g
    D) $40$ g
  2. If $\log_2(x)=5$, determine $x$.
    A) $10$
    B) $16$
    C) $25$
    D) $32$
  3. A bacteria colony triples every hour. Starting at $50$, determine the exact count after $3$ hours. (SPR)
  4. A model $Q(t) = Q_0 e^{-kt}$ has half-life $\ln 2 / k$. If $k = 0.1$, determine, to the nearest year, the half-life.
    A) $3$
    B) $5$
    C) $7$
    D) $10$

Level 3: Hard (Advanced)

  1. The equation $2 \cdot 3^{x} = 54$ has which exact solution?
    A) $x=2$
    B) $x=3$
    C) $x=\log_3 27$
    D) Both B and C
  2. A quantity grows at continuous rate $r = 0.08$ per year. Determine the doubling time in years (nearest integer).
    A) $7$
    B) $9$
    C) $12$
    D) $14$
  3. If $\ln x = 3$, determine, to $2$ decimal places, the value of $x$. (SPR)

Answer key

The answer to every question, with a brief explanation. Try the sheet first.

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