Interpreting the Slope and Vertical Intercept of a Linear Model

3.1.3 Interpreting the Slope and Vertical Intercept of a Linear Model

3.1.3 • Interpreting the Slope and Vertical Intercept of a Linear Model

Activity

The scatter plot shows the sale price of several food items, $y$ , and the cost of the ingredients used to produce those items, $x$ , as well as a line that models the data. The line is also represented by the equation $y = 3.48 x + 0.76$ .

Scatter plot showing a positive correlation between ingredient cost and sales price in dollars, with a blue trend line indicating that higher ingredient costs correspond to higher sales prices.

What is the predicted sale price of an item that has ingredients that cost $1.50? Be prepared to show your reasoning.

Compare your answer:

$5.98 from the equation $(3.48 \cdot 1.5 + 0.76 = 5.98)$ , or approximately $6 from the scatter plot graph

What is the predicted ingredient cost of an item that has a sale price of $7? Be prepared to show your reasoning.

Compare your answer:

$1.79 from the equation $(\frac{7 - 0.76}{3.48} = 1.79)$ , or approximately $1.75 from the scatter plot graph

What is the slope of the linear model?

Compare your answer:

3.48.

What does the slope value mean in this situation?

Compare your answer: Your answer may vary, but here is a sample. This means that for every extra dollar spent on ingredients, the sale price of the food item is expected to increase by about $3.48

What is the 𝑦-intercept of the linear model?

Compare your answer:

0.76

What does the value of the $y$ -intercept mean in this situation?

Compare your answer: Your answer may vary, but here is a sample. This means that a food item that is made with free ingredients would still be sold for approximately 76 cents.

Does the value of the $y$ -intercept make sense in this situation?

Compare your answer: Your answer may vary, but here are some samples.

This doesn’t seem to make sense because something that is free to make should not cost anything for the customer.
This might make sense since it would still cost some money to either run the machines or pay the people to make the item.

Video: Using Linear Models

Watch the following video to learn more about how to use linear models.

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Self Check

The scatter plot below shows the age (in years) and the price (in dollars) of used compact cars advertised in the local newspaper.

A SCATTER PLOT THAT SHOWS AGE IN YEARS ON THE X-AXIS AND PRICE IN DOLLARS ON THE Y-AXIS. THE LINE DRAWN DECREASES FROM LEFT TO RIGHT AND PASSES THROUGH THE POINTS (13, 6000) AND (7, 12000).

Nora drew a line to fit the data. What is the slope of her line, and what does it mean in the problem?

$-\frac1{1000}$ , for every year older the car is, it decreases $\frac1{1000}$ of its value the year before.
$1000$ , for every year older the car is, its price increases 1000 dollars.
$1000$ , for every year older the car is, its price decreases 1000 dollars.
$-1000$ , for every year older the car is, its price decreases 1000 dollars.

Additional Resources

Interpreting a Linear Model

The scatter plot below shows the height and speed of some of the world’s fastest roller coasters. A line has been drawn to show a good fit for the data.

A scatter plot that shows maximum height in feet on the x-axis and speed in miles per hour on the y-axis. The line drawn increases from left to right.

The equation of the line drawn is $S = 0.11 h + 60$ .

What does this slope mean in the problem?

For every foot taller the roller coaster is, the speed increases 0.11 miles per hour.

Try it

Interpreting a Linear Model

Eight students were asked to estimate their score on a 10-point quiz. Their estimated and actual scores are given in the table below. Plot the points. Then sketch a line that fits the data.

Predicted	Actual
6	6
7	7
7	8
8	8
7	9
9	10
10	10
10	9

Here are the points from the table plotted on a scatter plot, with the $x$ -axis being the predicted scores and the $y$ -axis being the actual scores.

A scatter plot that shows predicted scores on the x-axis and actual scores on the y-axis. The drawn best-fit line increases from left to right.

Tell what you would expect someone’s grade to actually be if they predicted their grade to be a 5 out of 10.

Compare your answer:

Using the linear model above, if a student predicted a 5 on the quiz, they should have an actual score of 6.

What does the slope mean in this problem?

Compare your answer:

The slope is the change in the actual grade point for each predicted grade point.