Question

1. a vine planted today has a height of 5 feet and grows 1 foot each month up the side of a brick building. is the height of the vine a function of the number of months? if so, is it a linear or a nonlinear function? explain your reasoning.

Explanation:

1. Function Check: For each number of months (input), there's exactly one height (output). So height is a function of months.
2. Linear Check: Let \( h \) = height (ft), \( m \) = months. The relationship is \( h = 5 + 1 \cdot m \), which fits \( y = mx + b \) (slope \( m = 1 \), y - intercept \( b = 5 \)), so it’s linear.

Answer:

Yes, the height of the vine is a function of the number of months, and it is a linear function.

• Function Reasoning: For any number of months \( m \), there is exactly one height \( h \) (e.g., \( m = 1 \) gives \( h = 6 \), \( m = 2 \) gives \( h = 7 \), etc.).
• Linear Reasoning: The height follows \( h = 5 + m \), which matches the linear function form \( y = mx + b \) (slope \( 1 \), y - intercept \( 5 \)).