Question

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Graphing Proportional Relationships Quick Check
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Question
Use the image to answer the 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.

Answer 1

To derive the equation of the line from the graph, we need to identify the proportional relationship between inches and feet.

From the points provided, we can observe the coordinates:
- (0, 0)
- (1, 12)
- (2, 24)
- (3, 36)

This suggests a linear relationship where the vertical axis (inches) increases as the horizontal axis (feet) increases. The relationship appears to be directly proportional, and we can express it in the form of \( y = kx \), where \( y \) is the output (inches), \( x \) is the input (feet), and \( k \) is the constant of proportionality.

To find \( k \), we can use one of the points from the graph. Let's use the point (1, 12):

1. Substitute \( x = 1 \) and \( y = 12 \) into the equation \( y = kx \):
\[
12 = k \cdot 1
\]

2. Therefore, \( k = 12 \).

Now we can write the equation of the line:
\[
y = 12x
\]

Thus, the equation of the line that represents the proportional relationship between inches and feet is:
\[
\text{Inches} = 12 \times \text{Feet}
\]