Question

Linear Functions Given Numerically A table of values for a linear function$f$is given.

1. Find the rate of change of$f$.
2. Express$f$in the form$f\left(x\right)=ax+b$.

Short answer

1. The rate of change of$f$ is $-3$.
2. The required function is $f\left(x\right)=-3x+2$.

Step-by-step solution

Part a. Step 1. Slope formula.

If the graph of a linear function passes through the two points$\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$, then the rate of change of the function is:

$m=\frac{y_2-y_1}{x_2-x_1}$

Part a. Step 2. Given information.

The table of values for a linear function$f$ is:

Part a. Step 3. Find the rate of change.

From the given table, it is clear that the linear function passes through the points$\left(-3,11\right)$ and $\left(0,2\right)$. So, the rate of change of the linear function$f$ is:

$$\begin{gathered} m=\frac{2-11}{0-(-3)} \\ =\frac{-9}{3} \\ =-3 \end{gathered}$$

Thus the rate of change of$f$ is $-3$.

Part b. Step 1. Definition of linear function.

In a linear function, each input value has a unique output value. In other words, for each point in domain, there is a unique value in the range of function. It can be written as:

$f\left(x\right)=ax+b$ … (i)

Where$a$ is the rate of change (slope) and$b$ is the initial value ($y$-intercept), and both are real numbers.

Part b. Step 2. Given information.

From the above part, it is clear that the rate of change of$f$ is $-3$, so $a=-3$.

The linear function$f$ passes through the point $\left(0,2\right)$. It means the $y$-intercept of the function$f$ is $2$, so $b=2$.

Part b. Step 3. Write the function.

Substitute$a=-3$ and$b=2$ in (i) to find the function $f$.

$\begin{array}{l}f\left(x\right)=\left(-3\right)x+\left(2\right)\\ f\left(x\right)=-3x+2\end{array}$

Thus, the required function is $f\left(x\right)=-3x+2$.