Question

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

Short answer

As the value of b increases, the graph of f is shifted upward.

The rate of change of f remains same

Step-by-step solution

Step 1. Given Information.

Given that, the function is

$f\left(x\right)=x+b$

Step 2. Graph of the function fx=x+b for b=12,b=1,and b=2.

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

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

From the graph, we can see that,

As the value of b increases, the graph of f is shifted upward and there is no change in the steepness.

Or in other words,

With the increase in the y-intercept of the line (or $f\left(x\right)=x+b$), the graph of f is shifted upward and there is no change in the steepness.

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 remains same i.e., there is no change in the rate of change of f.