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

Vertex form:

`G(x)=-(12)/(25)(x-1)^2+9`

Standard form:

`y=-(12)/(25)x^2+(24)/(25)x+(213)/(25)`

Step-by-step 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=-(12)/(25) \end{gathered}`

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

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