Question

Slope and linear functions: If f is a linear function with slope $-3$such that $f\left(2\right)=1,$find the following, without first finding an equation for $f\left(x\right).$

$$\begin{gathered} f\left(4\right) \\ f\left(7\right) \\ f(-2) \end{gathered}$$

Short answer

Function values are following.

$$\begin{gathered} f\left(4\right)=-8 \\ f\left(7\right)=-17 \\ f(-2)=10 \end{gathered}$$

Step-by-step solution

Step 1. Given information.

The given slope of the linear function is $m=-3.$

The given function value is$f\left(2\right)=1.$

Step 2. Value of f(4).

Substitute $y=f\left(4\right),x=4,m=-3,x_0=2,\&y_0=1$in standard point-slope form of a linear equation and solve for y.

$$\begin{gathered} y-y_0=m(x-x_0) \\ f\left(4\right)-1=-3(4-1) \\ f\left(4\right)-1=-9 \\ f\left(4\right)=-8 \end{gathered}$$

So function value is$f\left(4\right)=-8$

Step 3. Value of f(7).

Substitute $y=f\left(7\right),x=7,m=-3,x_0=2,\&y_0=1$in standard point-slope form of a linear equation and solve for y.

$$\begin{gathered} y-y_0=m(x-x_0) \\ f\left(7\right)-1=-3(7-1) \\ f\left(7\right)-1=-18 \\ f\left(7\right)=-17 \end{gathered}$$

So function value is$f\left(7\right)=-17.$

Step 4. Value of f(-2).

Substitute $y=f(-2),x=-2,m=-3,x_0=2,\&y_0=1$in standard point-slope form of a linear equation and solve for y.

role="math" localid="1649800928489" $$\begin{gathered} y-y_0=m(x-x_0) \\ f(-2)-1=-3(-2-1) \\ f(-2)-1=9 \\ f(-2)=10 \end{gathered}$$

So function value is$f(-2)=10.$