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?
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.
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.

The correct answer is 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.

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facundo3141592 facundo3141592 · Answer 2 · 2021-10-08T15:19:25+02:00

We want to study the possible solutions of a system of equations where one equation is a parabola and the other equation is a line.

We will see that the correct option is 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."

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Here we have the system of equations:

y = (x + 3)^2
y = m*x + 9

Isabel claims that one of the solutions to the system must always be the vertex of the parabola.

Now, the solutions of the system are the points where both graphs intersect, so to find the solutions we can write:

(x + 3)^2 = y = m*x + 9

Then the solutions can be found by solving:

(x + 3)^2 = m*x + 9

Now, the vertex of the parabola is at x = 0, so to test Isabel claim, we can evaluate the above expression in x = 0 to get:

(0 + 3)^2 = m*0 + 9

3^2 = 9

9 = 9

This is true, so Isabel is correct, there will always be a solution and that solution is the vertex of the given parabola.

Then the correct option is:

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.

(Option B is incorrect because the y-intercept of line B is written incorrectly there).

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