Chapter 8, Problem 19CT

To determine

To write:

A linear function that satisfies the given conditions.

Answer to Problem 19CT

The function $g(x)=-\frac{1}{5}x-6$ satisfies the given conditions.

Explanation of Solution

Given:

  $g(0)=-6,g(15)=-9$

Concepts used:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1},\text{where,}\\ m=\text{Slope of line,}\\ y_2-y_1=\text{Difference between two }y\text{-coordinates}\\ x_2-x_1=\text{Difference between two }x\text{-coordinates of same }y-\text{coordinates}\end{array}$

The equation of a line in slope-intercept form is $y=mx+b$ , where m represents slope of line and b represents the y-intercept.

Calculation:

We have been given two points on a line that are $(0,-6)\text{and }(15,-9)$ .

Now we will find the slope of line passing through the given points using slope formula.

Let point $(0,-6)=(x_1,y_1)\text{and (15,}-\text{9)}=(x_2,y_2)$ . Upon substituting our given coordinates in slope formula, we will get:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{-9-(-6)}{15-0}\\ m=\frac{-9+6}{15}\\ m=\frac{-3}{15}\\ m=-\frac{1}{5}\end{array}$

To write our function, we will substitute the value of slope and the y-intercept $(0,-6)$ in slope-intercept form of equation as shown below:

  $\begin{array}{l}y=mx+b\left(\text{Slope-intercept form of equation}\right)\\ y=-\frac{1}{5}\cdot x+(-6)\left(\text{Substituting }m=-\frac{1}{5}\text{ and }b=-6\right)\\ y=-\frac{1}{5}x-6\\ g(x)=-\frac{1}{5}x-6\left(\text{Writing in function form}\right)\end{array}$

Therefore, the function $g(x)=-\frac{1}{5}x-6$ satisfies the given conditions.