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.
- Set equal: $x^2 - 5x + 6 = x - 3$.
- Move to one side: $x^2 - 6x + 9 = 0$.
- Factor the perfect square trinomial: $(x - 3)^2 = 0$.
- Solve: $x = 3$.
- Wait, there is only one intersection point, which means the (Full worked solution in mark scheme.)
- 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.
- Set equal: $2x^2 - 8x + 11 = mx + 3$.
- Group the $x$ terms together: $2x^2 - (8 + m)x + 8 = 0$.
- Apply $\Delta = 0$: $(-(8 + m))^2 - 4(2)(8) = 0$.
- Simplify: $(8 + m)^2 - 64 = 0 \implies (8 + m)^2 = 64$.
- Take the square root: $8 + m = \pm 8$.
- 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!
- Set equal: $-2x^2 + 9x + 14 = 5x + c$.
- Move to one side: $-2x^2 + 4x + (14 - c) = 0$.
- Identify $a = -2$ and $b = 4$.
- Apply the sum formula: $-\frac{b}{a} = -\frac{4}{-2} = 2$.
- Answer: $2$.
Practice Questions
Level 1: Easy (Foundational SAT)
- 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)
- 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 $x_2$?
- 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)
- 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) $20$
- 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)
- 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) $8$
- 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)
- 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) $18$
- 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
- 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)
- 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)
- 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) $13$
- How many real solutions exist for the system of equations $y = 2x^2 + 5$ and $y = -2x^2 - 5$?
- 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)
- 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)
- 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) $k > -\frac{1}{3}$
- 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) $40$
- 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)
- 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)
- 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 & Brief Explanations
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