Questions: A calculator is allowed on this question. Malachi has been tracking his height each year since he was 7 years old. This year he is going to turn 12. Which linear function best models the data? Based on the data shown below, how tall should Malachi expect to be at 12 years old? Age (x) Height, in inches (f(x)) 7 8 53.0 9 55.5 10 58.0

Solution Steps

To find the linear function that best models the data, we need to perform a linear regression on the given age and height data points. Once we have the linear function, we can use it to predict Malachi's height at 12 years old.

We are given the following data points for Malachi's age and height:

• Age 7: Height 53.0 inches
• Age 8: Height unknown
• Age 9: Height 55.5 inches
• Age 10: Height 58.0 inches

We need to find a linear function \( f(x) = mx + b \) that models this data and then use it to predict Malachi's height at age 12.

To find the slope \( m \) of the linear function, we use the known data points. We can use the points (9, 55.5) and (10, 58.0).

The formula for the slope \( m \) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Substituting the values: \[ m = \frac{58.0 - 55.5}{10 - 9} = \frac{2.5}{1} = 2.5 \]

We use one of the known points and the slope to find the y-intercept \( b \). Using the point (9, 55.5):

\[ f(x) = mx + b \implies 55.5 = 2.5 \cdot 9 + b \]

Solving for \( b \): \[ 55.5 = 22.5 + b \implies b = 55.5 - 22.5 = 33.0 \]

Now that we have \( m \) and \( b \), we can write the linear function: \[ f(x) = 2.5x + 33.0 \]

To find Malachi's height at age 12, we substitute \( x = 12 \) into the linear function: \[ f(12) = 2.5 \cdot 12 + 33.0 = 30.0 + 33.0 = 63.0 \]

Final Answer