# Power Functions and Variation on a Graphing Calculator

## Key Questions

• All you need to do is go to your $y =$ button, and type the function in as you would for any other function. You may need to use the "carrot" key (located below the "Clear" button on the upper right side of the calculator) a fair few times. Watch your parenthesis.

For example, if I wanted to graph $y = 3 \left({5}^{x}\right)$, then I would enter the following:

3, (, 5, ^, x, ).

In words:

Three, left bracket, carrot, x, right bracket.

As I mentioned, just watch your parenthesis, as they are probably what most people are likely to mess up. If you pay attention to this you should be good :)

• First, let's remember what a power function is. For any constant real number $p$ and real variable $x$, functions of the form $f \left(x\right) = {x}^{p}$ are called power functions of $x$.

What does it mean that a power function ${x}^{p}$ models a given set of ordered pairs? We know that, in general, the graph of a function $f$ is the set of ordered pairs $\left(x , y\right)$, such that $y = f \left(x\right)$.

So, a given set of ordered pairs modeled by a power function corresponds to a set of points contained in the graph of the power function.

The problem becomes that of finding the equation of the power function given that we know the coordinates of a number of points of its graph. This is solved by solving the resulting system of equations.

Example:

Consider a given set of ordered pairs: $\left\{\begin{matrix}1 & 2 \\ 2 & 5 \\ 3 & 10\end{matrix}\right\}$. This corresponds to the points $A \left(1 , 2\right)$, $B \left(2 , 5\right)$ and $C \left(3 , 10\right)$ contained in the graph of the power function $f \left(x\right) = {x}^{p}$.

So, we have the following equations:
$\left(1\right)$ ${1}^{p} = 1$
$\left(2\right)$ ${2}^{p} = 4$
$\left(3\right)$ ${3}^{p} = 9$

We see that $p = 2$, therefore the power function is $f \left(x\right) = {x}^{2}$

This is a simple illustration of the general idea. In practice, the problems can be more complex.

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