Chapter 4, Problem 1RE

a.

To determine

The slope and the y-intercept of the linear function $f\left(x\right)=2x-5$.

Answer to Problem 1RE

The slope and the y-intercept of the linear function $f\left(x\right)=2x-5$ are $\underset{\_}{2\text{ and }-\text{5}}$ respectively.

Explanation of Solution

Result used:

Slope of a function:

The slope of a function $f\left(x\right)=mx+b$ is given by m.

The y-intercept of a function:

The y-intercept of a function $f\left(x\right)=mx+b$ is given by b.

Calculation:

Compare the function $f\left(x\right)=2x-5$ with $f\left(x\right)=mx+b$ and obtain the values $m=2\text{ and }b=-5$.

Therefore, the slope and the y-intercept of the linear function $f\left(x\right)=2x-5$ are $\underset{\_}{2\text{ and }-\text{5}}$ respectively.

b.

To determine

The average rate of change of the function $f\left(x\right)=2x-5$.

Answer to Problem 1RE

The average rate of change of the function $f\left(x\right)=2x-5$ is $\underset{\_}{2}$.

Explanation of Solution

Result used:

The average rate of change of a linear function:

The average rate of change of a linear function $f\left(x\right)=mx+b$ denoted by $\frac{\Delta y}{\Delta x}$ is a constant and is given by the following formula.

$\frac{\Delta y}{\Delta x}=m\cdots \cdots \left(1\right)$

Calculation:

For the given function $m=2$.

Use equation (1) and obtain the average rate of change of the given function as follows.

$\frac{\Delta y}{\Delta x}=2$

Therefore, the average rate of change of the function $f\left(x\right)=2x-5$ is $\underset{\_}{2}$.

c.

To determine

To graph: The function $f\left(x\right)=2x-5$ along with labelled intercepts.

Explanation of Solution

Consider the function $f\left(x\right)=2x-5$.

Rewrite the above function as $y=2x-5$.

Substitute the value $y=0$ in the equation $y=2x-5$ and obtain the x-intercept as shown below.

$\begin{array}{l}2x-5=0\\ \Rightarrow 2x=5\\ \Rightarrow x=\frac{5}{2}\end{array}$

Hence, the x-intercept of the given function is $\left(\frac{5}{2},0\right)$.

From part a., the y-intercept of f is given by $y=-5$.

Therefore, the point that represents the y-intercept is $\left(0,-5\right)$.

Use the points $\left(\frac{5}{2},0\right)\text{ and }\left(0,-5\right)$ and sketch the graph of the function $f\left(x\right)=2x-5$ as shown below in Figure 1.

From Figure 1, it can be observed that the intercepts of the function $f\left(x\right)=2x-5$ are $\left(\frac{5}{2},0\right)\text{ and }\left(0,-5\right)$.

d.

To determine

Whether the function $f\left(x\right)=2x-5$ is increasing is increasing, decreasing or constant.

Answer to Problem 1RE

The function $f\left(x\right)=2x-5$ is $\underset{\_}{\text{increasing}}$.

Explanation of Solution

Result used:

A linear function $f\left(x\right)=mx+b$ with positive slope m is increasing over its domain.

Calculation:

The slope of the function $f\left(x\right)=2x-5$ is 2.

Since the value 2 is positive, the function $f\left(x\right)=2x-5$ is $\underset{\_}{\text{increasing}}$.