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

Short 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-by-step solution

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.