Question

Families of Linear Functions Graph$f\left(x\right)=ax\text{for}a=\frac{1}{2},a=1,\text{\hspace{0.33em}and\hspace{0.33em}}a=2$ all on the same set of axes. How does increasing the value of a affect the graph of f? What about the rate of change of f?

Short answer

As the value of a increases, the graph of f becomes steeper.

The rate of change of f is increasing.

Step-by-step solution

Step 1. Given Information.

Given that, the function is

$f\left(x\right)=ax$

Step 2. Graph of the function fx=ax for a=12,a=1,and a=2.

The Graph for the function for the given a value is as follows:

Step 3. To find how the increasing value of a affect the graph of f.

From the graph, we can see that,

As the value of a increases, the graph of f becomes steeper.

Or in other words,

With the increase in the slope of the line (or $f\left(x\right)=ax$), the graph of f becomes steeper.

Step 4. To find the rate of change of f.

With the help of Step 3, we can conclude that,

The rate of change of f is increasing.