Uses Of Linear Models With Examples

Key Concepts

• Using of slope to make the predictions.

• Using scatter plot to make predictions.

• Interpretation of slope and y-intercept.

Introduction:

Making predictions:  

Scatter plots can be used to make predictions about current or future trends. By determining the equation of a linear model, predictions of an outcome can be made. 

The slope: 

The slope of a linear function is the change in the dependent variable divided by the change in the independent variable, or change in y/change in x. 

The slope of a line with the equation y= mx+ b is m. 

The x-intercept and y-intercept: 

The x-intercept of a graph is the point where the graph crosses the x-axis. 

The y-intercept of a graph is the point where the graph crosses the y-axis. 

The y-intercept of a line with the equation y= mx+ b is b. 

4.3.1 Using of Slope to Make the Predictions 

Example 1: 

Make a scatter plot to represent the data. Draw a trend line and write an equation for the trend line. Use the equation to predict the time needed to travel 32 miles on a bicycle. 

Solution: 

Step 1  Draw a scatter plot. Use a straight edge to draw a trend line. Estimate two points on the line. 

Step 2  Stack, subtract and divide to find the slope. 

(22, 107) (5,27) = Change in y/Change in x  = 80/17 

≈ 5 

so let m = 5 

y = m x + b   Use slope intercept form and solve for b. 

27 = 5(5) + b Substitute 5 for m and (5, 27) for (x, y). 

27 = 25 + b Solve for b 

2 =   b 

Step 3  Predict the time needed to travel 32 miles.  

y = 5(32) + 2 Substitute 32 for x. 

y = 160 + 2 Multiply. 

y = 162 Add. 

The time needed to travel 32 miles is about 162 minutes. 

4.3.2 Using Scatter Plot to Make Predictions 

Example 2: 

The graph shows the number of gallons of water in a large tank as it is being filled. Based on the trend line, predict how long it will take to fill the tank with 210 gallons of water. 

Solution: 

Find the x-value for a y-value of 210. 

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It will take about 10 minutes to fill the tank with 210 gallons of water. 

4.3.3 Interpretation of Slope and y-Intercept 

Example 3: 

A top fuel dragster (a car built for drag racing) can travel ¼ mile in 4 seconds. The dragster’s distance over time is graphed on the next slide. The graph assumes a constant speed. Use the graph to complete problems 1 and 2. Then use what you know about slope-intercept form to answer the remaining questions. 

1. Find the slope shown in the graph. 

2. Find the y-intercept of the function shown in the graph. 

3. Write the algebraic equation of the line. 

Solution: 

1. Find the slope shown in the graph. 

The slope is

change in y change in x change in y change in x

, To calculate the slope, find any two points on the line.  

The graph shows that (0, 0) and (16, 1) are both points on the line. The formula to find the slope between two points (x₁, y₁) and (x₂, y₂) is y2 – y1 / x2 – x1. Substitute (0, 0) and (16,1) into the formula to find the slope.  

The slope between the two points (0, 0) and (16, 1) is 16. 

2. Find the y-intercept of the function shown in the graph. 

The y-intercept is the point at which the graph crosses the y-axis. The graph shows that the  
y-intercept is 0, or (0, 0). 

3. Write the algebraic equation of the line. 

The equation of a line can be written in the form y = mx + b, where m is the slope of the line and b is the y-intercept. The equation of the line is y = 16x + 0 or y = 16x. 

Exercise:

1. _________________can be used to make predictions about current or future trends.
2. Assuming the trend shown in the graph continues, use the equation of the trend line to predict average fuel consumption in miles per gallon in 2025.

• A smoothie café has the ingredients needed to make 50,000 smoothies on a day when the high temperature is expected to reach 80 ^oF. Should the café employees expect to have enough ingredients for the day’s smoothie sales? Explain.

• Harry is in charge of selling tickets for the school jazz concert at $2.00 for students and $4.00 for adults. She hopes that the total ticket sales will be about $600 in order to cover expenses and make a modest profit. Write a linear equation in standard form to model this situation.
• Find the x and y intercepts of 4x + 3y = 12.
• The graph shows a family’s grocery expenses based on the number of children in the family.

• Using the slope, predict the difference in the amount spent on groceries between a family with five children and a family with two children.

• The scatter plot shows a hiker’s elevation above sea level over time. The equation of the trend line shown is y = 8.77x + 686. To the nearest whole number, predict what the hiker’s elevation will be after 145 minutes.

• The scatter plot shows the number of people at a fair based on the outside temperature. How many fewer people can be predicted to be at the fair on a 100 °F day than on a 75 °F day?

• If x represents the number of years since 2000 and y represents the gas price, predict what the difference between the gas prices in 2013 and 2001 is? Round to the nearest hundredth.

• If x represents the number of months since the beginning of 2016, and y represents the total precipitation to date, then predict the amount of precipitation received between the end of March and the end of June.

Concept Map:

What have we learned:

• Use the slope and y-intercept to make a prediction.

• Use a scatter plot to make predictions.

• Interpretation of slope and y-intercept.