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