Question

what is the function rule for the linear function in this table?
slope-intercept form
y = mx + b
input (x) | 0 | 1 | 2 | 3
output (y) | 2 | 7 | 12 | 17

Explanation:

Step1: Find the slope \( m \)

The slope \( m \) of a linear function is calculated by the formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Using the points \( (0, 2) \) and \( (1, 7) \), we have \( m=\frac{7 - 2}{1 - 0}=\frac{5}{1} = 5 \).

Step2: Find the y - intercept \( b \)

The slope - intercept form is \( y=mx + b \). When \( x = 0 \), \( y=b \). From the table, when \( x = 0 \), \( y = 2 \), so \( b = 2 \).

Step3: Write the function rule

Substitute \( m = 5 \) and \( b = 2 \) into the slope - intercept form \( y=mx + b \). We get \( y = 5x+2 \).

Answer:

\( y = 5x + 2 \)