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) 3 5 7 9

Explanation:

Step1: Find the slope \( m \)

The slope \( m \) of a line is calculated by the formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Using the points \( (0, 3) \) and \( (1, 5) \), we have \( m=\frac{5 - 3}{1 - 0}=\frac{2}{1} = 2 \).

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

The slope - intercept form of a line is \( y=mx + b \). When \( x = 0 \), from the table, \( y = 3 \). Substituting \( x = 0 \), \( y = 3 \) and \( m = 2 \) into \( y=mx + b \), we get \( 3=2\times0 + b \), so \( b = 3 \).

Step3: Write the function rule

Substitute \( m = 2 \) and \( b = 3 \) into the slope - intercept form \( y=mx + b \). We get \( y = 2x+3 \).

Answer:

The function rule for the linear function is \( y = 2x + 3 \)