Question

The range of y = - 32x ^ 2 + 90x + 3

Answer 1

Given:

The given function is:

`y=-32x^2+90x+3`

To find:

The range of the given function.

Solution:

We have,

`y=-32x^2+90x+3`

It is a quadratic function because the highest power of the variable x is 2.

Here, the leading coefficient is -32 which is negative. So, the graph of the given function is a downward parabola.

If a quadratic function is
`f(x)=ax^2+bx+c`, then the vertex of the quadratic function is:

`Vertex=\left((-b)/(2a),f((-b)/(2a))\right)`

In the given function,
`a=-32,b=90,c=3`.

`(-b)/(2a)=(-90)/(2(-32))`

`(-b)/(2a)=(-90)/(-64)`

`(-b)/(2a)=(45)/(32)`

The value of the given function at
`x=(45)/(32)` is:

`y=-32((45)/(32))^2+90((45)/(32))+3`

`y=(2121)/(32)`

The vertex of the given downward parabola is
`\left((45)/(32),(2121)/(32)\right)`. It means the maximum value of the function is
`y=(2121)/(32)`. So,

`Range=\left\y\leq (2121)/(32)\right\`

`Range=\left(-\infty, (2121)/(32)\right ]`

Therefore, the range of the given function is
`\left (-\infty, (2121)/(32)\right ]`.