Use the image to answer the question.

Question

An illustration shows a graph labeled Inches per Feet. The horizontal axis is labeled feet and ranges from 0 to 4 in increments of 1. The vertical axis is labeled Inches and ranges from 0 to 40 in increments of 4. An upward slanting line segment connects points plotted at left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 12 right parenthesis, left parenthesis 2 comma 24 right parenthesis, and left parenthesis 3 comma 36 right parenthesis.

The proportional relationship between inches and feet is displayed on the graph. Derive the equation of the line through the origin.

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Answer 1

The relationship displayed in the graph is linear and proportional, meaning that the relationship between the two variables can be described by a linear equation of the form \( y = mx \), where \( m \) is the slope.

To derive the equation, we can find the slope \( m \) by using two points from the graph. Let's use the points \( (1, 12) \) and \( (0, 0) \) (the origin).

The slope \( m \) is calculated as follows:

\[ m = \frac{\Delta y}{\Delta x} = \frac{12 - 0}{1 - 0} = \frac{12}{1} = 12 \]

Since the line goes through the origin (0, 0), we can write the equation of the line as:

\[ y = 12x \]

This means that for every foot, there are 12 inches. Therefore, the equation representing the relationship between inches and feet is:

\[ \text{Inches} = 12 \times \text{Feet} \]