Chapter 8.7, Problem 9C

To determine

To write:

A linear function that satisfies the given conditions.

Answer to Problem 9C

The function $g(x)=-\frac{7}{2}x-5$ satisfies the given conditions.

Explanation of Solution

Given:

  $g(-6)=16,g(0)=-5$

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 $(-6,16)\text{and }(0,-5)$ .

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

Let point $(-6,16)=(x_1,y_1)\text{and }(0,-5)=(x_2,y_2)$ . Upon substituting our given coordinates in slope formula, we will get:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{-5-16}{0-(-6)}\\ m=\frac{-21}{0+6}\\ m=-\frac{7}{2}\end{array}$

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

  $\begin{array}{l}y=mx+b\left(\text{Slope-intercept form of equation}\right)\\ y=-\frac{7}{2}\cdot x+(-5)\left(\text{Substituting }m=-\frac{7}{2}\text{ and }b=-5\right)\\ g(x)=-\frac{7}{2}x-5\left(\text{Writing in function form}\right)\end{array}$

Therefore, the function $g(x)=-\frac{7}{2}x-5$ satisfies the given conditions.