Question

for exercises 7-9, write an equation that matches each table of values.
7.

input, x	0	1	2	3
output, y	-1	2	5	8

Explanation:

Step1: Identify the type of function

We check if the function is linear by looking at the rate of change (slope). The slope \( m \) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \).

For \( (0, - 1) \) and \( (1,2) \): \( m=\frac{2-(-1)}{1 - 0}=\frac{3}{1} = 3 \)

For \( (1,2) \) and \( (2,5) \): \( m=\frac{5 - 2}{2 - 1}=\frac{3}{1}=3 \)

For \( (2,5) \) and \( (3,8) \): \( m=\frac{8 - 5}{3 - 2}=\frac{3}{1}=3 \)

Since the slope is constant (\( m = 3 \)), it is a linear function.

Step2: Find the y - intercept

The general form of a linear equation is \( y=mx + b \), where \( b \) is the y - intercept (the value of \( y \) when \( x = 0 \)).

When \( x = 0 \), \( y=-1 \). So, substituting \( x = 0 \), \( y=-1 \) and \( m = 3 \) into \( y=mx + b \):

\( -1=3(0)+b \)

\( b=-1 \)

Step3: Write the equation

Substitute \( m = 3 \) and \( b=-1 \) into the slope - intercept form \( y=mx + b \).

We get \( y = 3x-1 \)

Answer:

\( y = 3x-1 \)