Chapter 1, Problem 68RE

(a)

To determine

The linear function f.

Answer to Problem 68RE

The linear function is $\frac{-3}{4}x-5$

Explanation of Solution

Given information:

Linear function

  $f\left(0\right)=-5,f\left(4\right)=-8$

Formula used:

The general equation of the linear function is $f\left(x\right)=ax+b$

Calculation:

The general equation of the linear function as below,

  $f\left(x\right)=ax+b$

Where, indicated function values are $f\left(a\right)=c_1\text{and}f\left(b\right)=c_2$

Consider the following indicated function values as below,

  $f\left(0\right)=-5\text{and }f\left(4\right)=-8$

The evaluation of linear function by above rule,

  $f\left(x\right)=ax+b$

Put, $f\left(0\right)=-5\text{and }f\left(4\right)=-8$

  $\begin{array}{l}-5=a\times 0+b\\ b=-5\end{array}$

Also

  $\begin{array}{l}-8=4a+b\\ 4a-5=-8\\ a=-\frac{3}{4}\\ 4a-5=-8\\ a=\frac{-3}{4}\end{array}$

Therefore

  $f\left(x\right)=\frac{-3}{4}x-5$

Thus the linear function is $\frac{-3}{4}x-5$

Conclusion:

The linear function is $\frac{-3}{4}x-5$

(b)

To determine

The graph of the function.

Answer to Problem 68RE

The graph is decreasing on the negative axis.

Explanation of Solution

Given information:

Linear function

  $f\left(0\right)=-5,f\left(4\right)=-8$

Formula used:

  $y=mx+b$

Calculation:

Graph the function

A linear function of x

  $y=mx+b$

Where m is a slope and (0, b) is y intercept represents the line.

Compare the function with this general equation, it signifies

  $m=-\frac{3}{4},b=-5$

To sketch the line, initially plot y intercept $\left(0,-5\right)$ . As slope is $-\frac{3}{4}$ , move 4 units to the right and 3 units downward. The second point thus reached is $\left(4,-8\right)$

Plot it and draw line passing through these two points as shown below

The graph of the linear function is,

  

Conclusion:

The graph is decreasing on the negative axis.