Question

a table of values of a linear function is shown below. find the output when the input is n. type your answer in the space provided.
input \t output
1 \t 10
2 \t 13
3 \t 16
4 \t 19
n \t blank

Explanation:

Step1: Find the slope (rate of change)

The slope \( m \) of a linear function between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Using the first two points \((1, 10)\) and \((2, 13)\), we have \( m=\frac{13 - 10}{2 - 1}=\frac{3}{1} = 3 \).

Step2: Find the y - intercept (b)

The equation of a linear function is \( y=mx + b \). Using the point \((1, 10)\) and \( m = 3 \), we substitute into the equation: \( 10=3(1)+b \). Solving for \( b \), we get \( 10 = 3 + b\), so \( b=10 - 3=7 \).

Step3: Write the linear function

The linear function is \( y = 3x+7 \). Here, the input is \( n \) (which is \( x \)) and the output is \( y \). So the output when the input is \( n \) is \( 3n + 7 \).

Answer:

\( 3n + 7 \)