Question

is the relationship a linear function? yes no for the table to the right, represent the relationship using words, an equation, and a graph. then determine whether the relationship is a linear function. x 0 1 2 3 y 2 5 8 11

Explanation:

Step1: Recall linear - function property

A linear function has a constant rate of change. We can find the rate of change (slope) using the formula $m=\frac{y_2 - y_1}{x_2 - x_1}$ for two points $(x_1,y_1)$ and $(x_2,y_2)$.
Let's take the points from the table: $(0,2)$ and $(1,5)$.
$m_1=\frac{5 - 2}{1 - 0}=\frac{3}{1}=3$.

Step2: Check other points

Take the points $(1,5)$ and $(2,8)$.
$m_2=\frac{8 - 5}{2 - 1}=\frac{3}{1}=3$.
Take the points $(2,8)$ and $(3,11)$.
$m_3=\frac{11 - 8}{3 - 2}=\frac{3}{1}=3$.
Since the rate of change (slope) between any two - points is constant ($m = 3$), the relationship is a linear function.
The equation of a line in slope - intercept form is $y=mx + b$, where $m$ is the slope and $b$ is the y - intercept. From the point $(0,2)$, we know that $b = 2$ and $m = 3$, so the equation is $y=3x + 2$.

Answer:

Yes