Q. 16 - Precalculus Enhanced with Graphing Utilities

Question

If $p\left(x\right)=-x+6,$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 $-1$and y- intercept is $6.$

(b) Graph of function is: - 

(C)The average rate of change of function is$-1.$

(d)The slope of given linear function is negative so, function is decreasing.  

Step 1 Given information

Given that the linear function is $p\left(x\right)=-x+6.$

Part(a) Step 2 Solution

Here, the linear function is $p\left(x\right)=-x+6.$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 $-1$and y- intercept $6.$

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 $p\left(x\right)=-x+6$, the average rate of change is $-1.$

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 $m,$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 $p\left(x\right)=-x+6$, slope is negative so linear function is decreasing.