Chapter 3, Problem 2RE

Solution

(a)

To find: the slope and y-intercept of each linear function.

Slope $m=-4$ and $y$ intercept $b=(0,7)$

Given:

  $g(x)=-4x+7$

Calculation:

The slope intercept form of a line as $y=mx+b$ .

Where the slope of the line is $m$

And $y$ intercept is $b$

So on comparing the function $f(x)=-4x+7$ with slope intercept form.

Slope $m=-4$ and $y$ intercept $b=(0,7)$

Conclusion:

Therefore, the slope $m=-4$ and $y$ intercept $b=(0,7)$

(b)

To find:the average rate of change of each function

The rate of change is -4

Calculation:

For calculating average rate of change of given function, it is needed two points for it. Y intercept is one point (0.7) and

For second point

Let $x=1$

  $f(1)=-4\times 1+7$

  $\begin{array}{l}=-4+7\hfill \\ =3\hfill \end{array}$

So second point in $(x,y)$ form is (1,3) .

Now as rate of change $\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$

Here $x_1=0,y_1=7,x_2=1,y_2=3$

So rate of change $=\frac{3-7}{1-0}$

  $=\frac{-4}{1}$= -4

Conclusion:

Therefore, the rate of change =-4

(c)

To graph: the given function.

Calculation:

The line joining two points (0,7) and (1,3) is shown as below where (1.75,0) is x intercept.

  

Conclusion:

Thus, the given function is drawn.

(d)

To determine:whether the function is increasing decreasing or constant.

Thus function is decreasing in the interval $[-\infty ,+\infty ]$

Calculation:

As the slope of the given function is -4 that is negative, that means it is decreasing function as if slope is negative that means this line makes a obtuse angle with positive side of $x$ axis. Thus function is decreasing in the interval $[-\infty ,+\infty ]$

Conclusion:

Thus function is decreasing in the interval $[-\infty ,+\infty ]$