Chapter 3, Problem 1CT

(a)

To determine

To find: the slope and y -intercept

Answer to Problem 1CT

Slope $m=-4$

y -intercept $b=3$

Explanation of Solution

Given:

f(x)=-4 x+3

Calculation:

Recall that the slope-intercept form of a line is $y=mx+b$

Where the slope of the line is $m$ and $y$ intercept is $b$ .

Compare the given function with the slope-intercept form.

Slope $m=-4$

y -intercept $b=3$

Conclusion:

Therefore, the slope $m=-4$ and y -intercept $b=3$

(b)

To determine

To find: the average rate of change of f

Answer to Problem 1CT

The rate of change =-4

Explanation of Solution

Calculation:

For calculating average rate of change of given function, we need any two points for it. Also $y$

-intercept is our one point and for second point

Let $x=1$

  $f(1)=-4\times 1+3$

  $\begin{array}{l}=-4+3\hfill \\ =-1\hfill \end{array}$

So, the second coordinate in $(x,y)$ form is (1,-1)

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

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

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

  $=\frac{-4}{1}=-4$

Conclusion:

The rate of change =-4

(c)

To determine

whether f is increasing, decreasing, or constant.

Answer to Problem 1CT

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

Explanation of Solution

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 its line makes an 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 $[-\infty ,\infty ]$ .

(d)

To determine

To graph: the function f.

Answer to Problem 1CT

  

Explanation of Solution

Calculation:

Sketch the graph of line joining two points (0,3) and (1,-1) where (0.75,0) is the $x$ -intercept

Conclusion:

Thus, the required graph is drawn.