Q. 17 - Precalculus Enhanced with Graphing Utilities

Question

If $f\left(x\right)=\frac{1}{4}x-3,$then

(a) Determine the slope and y-intercept of each function.

(b) Use the slope and y-intercept to graph the linear function.

(c) Determine the average rate of change of each function.

(d) Determine whether the linear function is increasing, decreasing, or constant 

Step-by-Step Solution

Answer

(a) The Slope of the given function is $\frac{1}{4}$and y- intercept is $-3.$

(b) Graph of function is: -  

(C)The average rate of change of function is $\frac{1}{4}.$

(d)The slope of given linear function is positive so, function is increasing.  

Step 1 Given Information

Given that the linear function is $f\left(x\right)=\frac{1}{4}x-3.$

Part(a) Step 2 Solution

Here, the linear function is $f\left(x\right)=\frac{1}{4}x-3.$We know that in a linear function$f\left(x\right)=mx+b$, $m$is the slope and  $b$ is y- intercept.

So, in the given function slope is $\frac{1}{4}$and y- intercept is $-3.$

Part(b) Step 3 Solution

The graph of the given function is 

Part (c) Step 4 Solution

Linear functions have a constant average rate of change. That is, the average rate of change of a linear function $f\left(x\right)=mx+b$ is $\frac{∆y}{∆x}=m.$

So, in the given linear function $f\left(x\right)=\frac{1}{4}x-3$, the average rate of change is $\frac{1}{4}.$

Part(d) Step 5 solution

Since, we know that the linear function $f\left(x\right)=mx+b,$is increasing over its domain if its slope $m,$is positive. It is decreasing over its domain if its slope $m,$is negative. It is constant over its domain if its slope $m,$ is zero.

Here, in given linear function $f\left(x\right)=\frac{1}{4}x-3,$ slope is positive so linear function is increasing.