Question

parabola a can be represented using the equation \\((x + 3)^2 = y\\), while line b can be represented using the equation \\(y = mx + 9\\). isabel claims one solution to the system of two equations must always be the vertex of parabola a. which best describes the reasonableness of her claim?

isabel is correct because the y-intercept of line b is \\((0, 9)\\) and the value of \\(y\\) when \\(x = 0\\) in parabola a is 9.
isabel is correct because the y-intercept of line b is \\((9, 0)\\) and the value of \\(x\\) when \\(y = 0\\) in parabola a is 9.
isabel is incorrect because the point of intersection between line b and parabola a that can be determined is the y-intercept of the equations, not the vertex.
isabel is incorrect because the point of intersection between line b and parabola a that can be determined is the x-intercept of the equations, not the vertex.

Explanation:

Find the vertex of Parabola A and check if it lies on Line B

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Since the vertex \((-3, 0)\) is only a solution when \(m = 3\), it is not always a solution for any real number \(m\). Thus, Isabel's claim is incorrect.

Find the guaranteed intersection point of the two equations

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Match with the correct multiple-choice option

The third option states: "Isabel is incorrect because the point of intersection between line B and parabola A that can be determined is the y-intercept of the equations, not the vertex."

Answer:

• (A) Isabel is correct because the y-intercept of line B is (0, 9) and the value of y when x = 0 in parabola A is 9
• (B) Isabel is correct because the y-intercept of line B is (9, 0) and the value of x when y = 0 in parabola A is 9
• (C) Isabel is incorrect because the point of intersection between line B and parabola A that can be determined is the y-intercept of the equations, not the vertex (Correct answer)
• (D) Isabel is incorrect because the point of intersection between line B and parabola A that can be determined is the x-intercept of the equations, not the vertex