Question

What is the range of the graph of `y = 5(x – 2)^2 + 7`?

Answer 1

`color(blue)(y in[7,oo)`

Explanation:

Notice `y=5(x-2)^2+7` is in the vertex form of a quadratic:

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

Where:

`bba` is the coefficient of `x^2`, `bbh` is the axis of symmetry and `bbk` is the maximum/minimum value of the function.

If:

`a>0` then the parabola is of the form `uuu` and `k` is a minimum value.

In example:

`5>0`

`k=7`

so `k` is a minimum value.

We now see what happens as `x->+-oo`:

as `x->oocolor(white)(88888)`, `5(x-2)^2+7->oo`

as `x->-oocolor(white)(888)`, `5(x-2)^2+7->oo`

So the range of the function in interval notation is:

`y in[7,oo)`

This is confirmed by the graph of `y=5(x-2)^2+7`

graph{y=5(x-2)^2+7 [-10, 10, -5, 41.6]}