Question

The following graph represents the hourly commission of a home appliance salesperson, y, based on the number of salespeople that are

working the showroom floor, x. The graph of the hourly commission of the salesperson is a quadratic function. What is the equation for
the function in standard form if the vertex is 6.5,42.25 and the origin is the only other known point?

Answer 1

We have been given graph of a downward opening parabola with vertex at point
`(6.5,42.25)`. We are asked to write equation of the parabola in standard form.

We know that equation of parabola in standard form is
`f(x)=ax^2+bx+c`.

We will write our equation in vertex form and then convert it into standard form.

Vertex for of parabola is
`y=a(x-h)^2+k`, where point (h,k) represents vertex of parabola and a represents leading coefficient.

Since our parabola is downward opening so leading coefficient will be negative.

Upon substituting coordinates of vertex and point (0,0) in vertex form, we will get:

`0=-a(0-6.5)^2+42.25`

`0=-a(42.25)+42.25`

`-42.25=-a(42.25)+42.25-42.25`

`-42.25=-42.25a`

`a=1` Divide both sides by
`-42.25`

So our equation in vertex form would be
`f(x)=-(x-6.5)^2+42.25`.

Let us convert it in standard from.

`f(x)=-(x^2-13x+42.25)+42.25`

`f(x)=-x^2+13x-42.25+42.25`

`f(x)=-x^2+13x`

Therefore, the equation of function is standard form would be
`f(x)=-x^2+13x`.