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

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Answer 1 · 2018-05-09T19:00:16

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

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]}

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