Q. 13 - Precalculus Enhanced with Graphing Utilities

Question

If $f\left(x\right)=2x+3$, 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$2$ and y- intercept is$3.$

(b) Graph of function is: - 

(C)The average rate of change of function is $2.$

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

Step 1 Given Information

Given that the linear function is $f\left(x\right)=2x+3.$

Part(a) Step 2 Solution

Here, the linear function is$f\left(x\right)=2x+3.$ We know that in a linear function $y=mx+b$, $m$ is the slope and $b,$ is y- intercept.

So, in the given function slope is $2$ and y- intercept $3.$

Part(b) Step 3 Solution

The graph of 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 $f\left(x\right)=2x+3$ the average rate of change is $2.$

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 $f\left(x\right)=2x+3$, slope is positive so linear function is increasing.