Chapter 4.2, Problem 26E

To determine

To calculate: To write the linear function

Answer to Problem 26E

The linear function is $f\left(x\right)=-\frac{3}{2}x-\frac{7}{2}$

Explanation of Solution

Given: $f\left(-9\right)=10;f\left(-1\right)=-2$

Formula Used:

Equation of line in slope intercept form is given as:

  $y=mx+c$ , where $\begin{array}{l}m=slope\\ c=y-intercept\end{array}$

Also, equation of line passing through two points is given as

  $y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)$

Calculation:

Given that values of linear function as follows:

  $f\left(-9\right)=10;f\left(-1\right)=-2$

This means that at $x=-9$ , $y=10$ and $x=-1$ , $y=-2$

Equation of function is found by finding the equation of line passing through $\left(-9,10\right)$ and $\left(-1,-2\right)$

Thus, equation of line is calculated as follows:

  $y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)$

Substituting the values in above equation:

  $\begin{array}{l}y-\left(-2\right)=\frac{-2-10}{-1-\left(-9\right)}\left(x-\left(-1\right)\right)\\ y+2=\frac{-2-10}{-1+9}\left(x+1\right)\\ y+2=\frac{-12}{8}\left(x+1\right)\\ y+2=\frac{-3}{2}\left(x+1\right)\\ 2y+4=-3x-3\\ 2y=-3x-7\\ y=-\frac{3}{2}x-\frac{7}{2}\end{array}$

Conclusion:

Hence, the linear function is $f\left(x\right)=-\frac{3}{2}x-\frac{7}{2}$