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 $\frac{3}{2}$.
2. The required function is $f\left(x\right)=\frac{3}{2}x+7$.

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$\left(0,7\right)$ and $\left(2,10\right)$. So, the rate of change of the linear function$f$ is:

role="math" localid="1646818682198" $$\begin{gathered} m=\frac{10-7}{2-0} \\ =\frac{3}{2} \end{gathered}$$

Thus the rate of change of$f$ is $\frac{3}{2}$.

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 $\frac{3}{2}$, so $a=\frac{3}{2}$.

The linear function$f$ passes through the point $\left(0,7\right)$. It means the $y$-intercept of the function$f$ is $7$, so $b=7$.

Part b. Step 3. Write the function.

Substitute$a=\frac{3}{2}$ and$b=7$ in (i) to find the function $f$.

$\begin{array}{l}f\left(x\right)=\left(\frac{3}{2}\right)x+\left(7\right)\\ f\left(x\right)=\frac{3}{2}x+7\end{array}$

Thus, the required function is $f\left(x\right)=\frac{3}{2}x+7$.