Question

A linear function h models a relationship in which the dependent variable increases 1 unit for every 5 units the independent variable decreases. The value of the function at 0 is 3.

Answer 1

To model this linear relationship with a linear function \( h \), we need to write it in the form of \( h(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept (the value of \( h(x) \) when \( x = 0 \)).

According to the information provided:

1. The dependent variable increases 1 unit for every 5 units the independent variable decreases. This means the slope (m) is -1/5 (since an increase in the dependent variable corresponds to a decrease in the independent variable). A decrease in the independent variable is represented by a negative slope.

2. The value of the function at 0 is 3. This means that when \( x = 0 \), \( h(x) = 3 \). Therefore, the y-intercept (b) is 3.

Putting this information together, we can write the function as:

\[ h(x) = -\frac{1}{5}x + 3 \]

Here, \(-\frac{1}{5}\) represents the rate of change (slope), and 3 is the value of the function when \( x = 0 \).