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 determines one solution to the system of two equations must always pass through the vertex of parabola a.
which best describes the reasonableness of her solution?
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 functions, not necessarily 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 functions, not necessarily the vertex.

the third option: Isabel is incorrect because the point of intersection between Line B and Parabola A that can be determined is the y-intercept of the functions, not necessarily the vertex.


Explanation:

1) The given equation of the parabola is y = (x + 3)², so that is in its vertex form: y = (x - h)² + k, where the vertex is (h,k).


Hence, the vertex of the parabola is the point ( -3,0).


2) The given equation of the line given is y = mx + 9, so that is the slope-intercept form, where the slope is m and the y-intercept is 9.


Hence, the y-intercept of the line is 9, that is the point (0,9).


3) The y-intercept of the parabola is determined doing x = 0 in its equation: => y = (0 + 3)² = 3² = 9. That is the point (0,9).


4) As you see, the point (0,9) is necessarily a solution of the system, but not the vertex of the parabola, (-3,0). The line might intercept or not the parabola at the vertex (-3,0) depending of the slope m.

The answer is the third option.

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