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

May 9, 2018

#### Answer:

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

#### Explanation:

Notice $y = 5 {\left(x - 2\right)}^{2} + 7$ is in the vertex form of a quadratic:

$y = a {\left(x - h\right)}^{2} + k$

Where:

$\boldsymbol{a}$ is the coefficient of ${x}^{2}$, $\boldsymbol{h}$ is the axis of symmetry and $\boldsymbol{k}$ is the maximum/minimum value of the function.

If:

$a > 0$ then the parabola is of the form $\bigcup$ and $k$ is a minimum value.

In example:

$5 > 0$

$k = 7$

so $k$ is a minimum value.

We now see what happens as $x \to \pm \infty$:

as $x \to \infty \textcolor{w h i t e}{88888}$, $5 {\left(x - 2\right)}^{2} + 7 \to \infty$

as $x \to - \infty \textcolor{w h i t e}{888}$, $5 {\left(x - 2\right)}^{2} + 7 \to \infty$

So the range of the function in interval notation is:

$y \in \left[7 , \infty\right)$

This is confirmed by the graph of $y = 5 {\left(x - 2\right)}^{2} + 7$

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