Answer:
The correct option is C.
Step-by-step explanation:
The vertex form of a parabola is
[tex]y=a(x-h)^2+k[/tex] .... (1)
Where, (h,k) is vertex.
From the given figure it is clear that the vertex of the parabola is at (-2.5, -3.25) and the y-intercept is (0,3).
Substitute h=-2.5, k=-3.25, x=0 and y=3 in equation (1) to find the value of a.
[tex]3=a(0+2.5)^2-3.25[/tex]
[tex]3+3.25=6.25a[/tex]
[tex]6.25=6.25a[/tex]
Divide both sides by 6.25.
[tex]1=a[/tex]
Substitute h=-2.5, k=-3.25 and a=1 in equation (1), to find the equation of parabola.
[tex]y=1(x+2.5)^2-3.25[/tex]
[tex]y=x^2+5x+6.25-3.25[/tex]
[tex]y=x^2+5x+3[/tex]
The equation of parabola is [tex]y=x^2+5x+3[/tex].
If a line passes through two points then the equation of line is
[tex]y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)[/tex]
The line passes through two points (-6,9) and (-5,3).
[tex]y-9=\frac{3-9}{-5+6}(x+6)[/tex]
[tex]y-9=\frac{-6}{1}(x+6)[/tex]
[tex]y-9=-6(x+6)[/tex]
[tex]y-9=-6x-36[/tex]
Add 9 on both the sides.
[tex]y=-6x-36+9[/tex]
[tex]y=-6x-27[/tex]
The equation of line is y=-6x-27.
Therefore the correct option is C.
