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 $x$.
- Substitute $y = 4$: $4 = x^2 - 5x + 10$.
- Set the equation to zero: $0 = x^2 - 5x + 6$.
- Factor the quadratic: $0 = (x - 2)(x - 3)$.
- The two points of intersection are at $x = 2$ and $x = 3$.
- The rightmost point has the larger $x$-value.
- 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.
- Set them equal: $x^2 + 3x - 5 = x + 3$.
- Move all terms to the left side: $x^2 + 2x - 8 = 0$.
- 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: $-b/a$.
- Sum of roots $= -\frac{2}{1} = -2$.
- 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 $0$.
- Set them equal: $x^2 - 6x + 14 = mx - 2$.
- Move all terms to the left side: $x^2 - 6x - mx + 16 = 0$.
- Group the $x$ terms to find $b$: $x^2 - (6 + m)x + 16 = 0$.
- Identify $a = 1$, $b = -(6 + m)$, $c = 16$.
- Set discriminant to zero: $(-(6+m))^2 - 4(1)(16) = 0 \implies (6+m)^2 - 64 = 0$.
- Solve for $m$: $(6+m)^2 = 64 \implies 6+m = \pm 8$.
- Therefore, $m = 2$ or $m = -14$. Since $m > 0$, $m = 2$.
- Answer: $2$.
Practice Questions
Level 1: Easy (Foundational SAT)
- 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)
- 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)
- 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
- 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?
- 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)
- 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?
- 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)
- 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)
- 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) $9$
- 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)
- $y = x^2 - 8x + 20$
$y = c$
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) - $y = x^2 + 4x + 9$
$y = mx$
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) - $y = -x^2 + 5x$
$y = 5$
How many distinct real solutions does the given system of equations have?
- A) Zero
- B) Exactly one
- C) Exactly two
- D) Infinitely many
- 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) $25$
- $y = x^2 + cx + 10$
$y = 2x + 1$
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)
- 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)
- $x^2 + y^2 = 25$
$y = x + 1$
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) - $y = 3x^2 - 7x + 4$
$y = 2x^2 - 2x - 2$
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) - $x = y^2 - 4y + 4$
$x = 4$
The graphs of the given equations intersect at two points. What is the sum of the $y$-coordinates of the intersection points?
- 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)
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