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SAT Digital Math · SL AA

Advanced Math: Quadratics & Parabolas

Targeted Practice Sheet 1

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)

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