Interpret Slope and y-intercept: Mini-Lesson Review

Mini Lesson Question

Question #3: Interpret Slope and $y$ -Intercept

The figure shows the graph of the equation used to convert temperature in Celsius to temperature in Fahrenheit: $F = \frac95C + 32$ .

$F = \frac95C + 32$ is used to convert temperatures, $C$ , on the Celsius scale to temperatures, $F$ , on the Fahrenheit scale. Identify and interpret the slope and $F$ -intercept.

The slope is $\frac95$ , which means that the temperature Fahrenheit ( $F$ ) increases 9 degrees when the temperature Celsius ( $C$ ) increases 5 degrees. The $F$ -intercept is -18, which means that when the temperature is 0° on the Celsius scale, it is 18° lower on the Fahrenheit scale.
The slope is 32, which means that for every 32 degrees Fahrenheit, it is 1 degree Celsius. The $F$ -intercept is $\frac95$ , which means that when the temperature is 0° on the Celsius scale, it is $\frac59°$ on the Fahrenheit scale.
The slope is $\frac95$ , which means that the temperature Celsius ( $C$ ) increases 9 degrees when the temperature Fahrenheit $(F)$ increases 5 degrees. The $F$ -intercept is 32, which means that when the temperature is 0° on the Celsius scale, it is 32° on the Fahrenheit scale.
The slope is $\frac95$ , which means that the temperature Fahrenheit $(F)$ increases $9$ degrees when the temperature Celsius ( $C$ ) increases 5 degrees. The $F$ -intercept is $32$ , which means that when the temperature is 0° on the Celsius scale, it is 32° on the Fahrenheit scale.

Interpret Slope and $y$ -Intercept

Many real-world applications are modeled by linear equations. Often, when a linear equation model uses real-world data, different letters are used for the variables, instead of using only $x$ and $y$ . The variable names remind us of what quantities are being measured.

The equation $T = \frac{1}{4} n + 40$ is used to estimate the temperature in degrees Fahrenheit, $T$ , based on the number of cricket chirps, $n$ , in one minute. The graph shows the relationship.

Graph of a line on the coordinate plane. The line intersects the y-axis at 40 and then slopes upward to the right. The x-axis extends from 0 to 48 with a scale of 4. The y-axis extends from 0 to 55 with a scale of 5.

What do the slope and $T$ -intercept represent?

Answer: The slope is $\frac{1}{4}$ and represents that for every 4 chirps per minute, the temperature increases by 1 degree. The $T$ - intercept means when there are zero chirps, the temperature is 40 degrees.

Try it

Try It: Interpret Slope and $y$ -intercepts

Loreen has a calligraphy business. The equation $C = 1.8 n + 35$ models the relation between her weekly cost, $C$ , in dollars and the number of wedding invitations, $n$ , that she writes. The graph shows the relationship.

Graph of a line on the coordinate plane. The line intersects the y-axis at 36 and then slopes upward to the right. The x-axis extends from 0 to 19 with a scale of 1. The y-axis extends from 0 to 60 with a scale of 4.

What do the slope and $C$ -intercept represent?

Here is how to interpret the slope and $C$ -intercept:

The slope is 1.8 and represents the cost increases by $1.80 for 1 invitation. The $C$ -intercept means the cost for zero invitations is $35.

Check Your Understanding

Sam drives a delivery van. The equation $C = 0.5 m + 60$ models the relation between his weekly cost, $C$ , in dollars and the number of miles, $m$ , that he drives. The graph shows the relationship.

Graph of a line on the coordinate plane. An orange line with an upward trend passes through black points at (60, 50) and (120, 100), indicating steady increase from left to right.

Find and interpret the meaning of the slope and $C$ -intercept.

The slope, 0.5, means that the weekly cost, $C$ , increases by $0.50 when the number of miles driven, $m$ , increases by 1. The $C$ -intercept means that when the number of miles driven is 0, the weekly cost is $60.
The slope, 0.5, means the weekly cost, $C$ , increases $0.50 a week. The $C$ -intercept is 60, which means that every mile costs $60.
The slope, 0.5, means that for every half mile, $m$ , the cost, $C$ , increases by 1. The $C$ -intercept means that when the number of miles driven is 0, the weekly cost is $60.
The slope, 0.5, means that the weekly cost, $C$ , decreases by $0.50 when the number of miles driven, $m$ , increases by 1. The $C$ -intercept means that when the number of miles driven is 60, the weekly cost is $0.

The slope, 0.5, means that the weekly cost, 𝐶 , increases by $0.50 when the number of miles driven, 𝑚 , increases by 1. The 𝐶-intercept means that when the number of miles driven is 0, the weekly cost is $60.

Check yourself: The slope is positive so the cost will increase per mile. The slope is the change in 𝐶 over the change in $m . \frac{0.5}{1}$ . The slope, 0.5, means that the weekly cost, 𝐶, increases by $0.50 when the number of miles driven, 𝑚, increases by 1. The 𝐶-intercept is the weekly cost when the miles are 0. Looking at the graph, this is the value where the line intersects the vertical axis. 60. The 𝐶-intercept means that when the number of miles driven is 0, the weekly cost is $60.

Videos: Interpreting Slope and $y$ -Intercept of Linear Equations

Khan Academy: Slope, $x$ -Intercept, and $y$ -Intercept Meaning in Context

Watch the following video to see how you can interpret slope, $x$ -intercept, and $y$ -intercept in context.

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Khan Academy: Slope and Intercept Meaning in Context

Watch this video to see how you can interpret slope and intercept meaning in context.

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Interpret Slope and y-intercept: Mini-Lesson Review

Mini Lesson Question

Question #3: Interpret Slope and y y -Intercept

Interpret Slope and y y -Intercept