Question

Use the vertex (h,k) and a point on the graph (x,k) to find the standard form of the equation of the quadratic function

Answer 1

Answer:

Vertex form:

$G(x)=-\frac{12}{25}(x-1)^2+9$

Standard form:

$y=-\frac{12}{25}x^2+\frac{24}{25}x+\frac{213}{25}$

Explanation:

A quadratic equation in standard form is generally given as;

$y=a(x-h)^2+k$

where (h, k) is the coordinate of the vertex.

Given the coordinate of the vertex as (1, 9) so we have that h = 1 and k = 9.

Also, given the coordinate of a point on the parabola as (-4, - 3), we have x = -4 and y = -3.

Let's go ahead and substitute the above values to the vertex equation and solve for a as seen below;

$\begin{gathered} -3=a(-4-1)^2+9 \\ -3=a(-5)^2+9 \\ -3=25a+9 \\ 25a=-3-9 \\ a=-\frac{12}{25} \end{gathered}$

So the vertex form of the quadratic equation can now be written as;

$y=-\frac{12}{25}(x-1)^2+9$

Let's go ahead and expand and simplify the above to have it in standard form;

$\begin{gathered} y=-\frac{12}{25}(x^2-2x+1)+9 \\ y=-\frac{12}{25}x^2+\frac{24}{25}x-\frac{12}{25}+9 \\ y=-\frac{12}{25}x^2+\frac{24}{25}x+(\frac{-12+225}{25}) \\ y=-\frac{12}{25}x^2+\frac{24}{25}x+\frac{213}{25} \end{gathered}$