solving problems given functions fitted to data/interpreting slope and y-intercept key terms: linear...

Solving Problems Given Functions Fitted to Data/Interpreting Slope and y-intercept

Key Terms:

Linear Fit

Slope

Scatter Plot

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4.6: Solving Problems Given Functions Fitted to Data

In this lesson we will be able to…1. Evaluate or solve a function via graph or x,y

table2. Solve a function algebraically by substituting a value for y and solving for x.3. Solve a function graphically by finding the point on the graph of the function with the known y-value, then finding the corresponding x-value of that point.4. To compare a data set and a function, plot the function on the same coordinate plane as the scatter plot of a data set. The graph of the function should approximate the shape of the scatter plot

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In this lesson we will be able to…

5. Graph a linear function by plotting two points and drawing a line through those two points

6. Graph an exponential function by plotting at least five points and connect those points with a curve.

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IntroductionWhen linear functions are used to model real-world relationships, the slope and y-intercept of the linear function can be interpreted in context. Recall that data in a scatter plot can be approximated using a linear fit, or linear function that models real-world relationships. A linear fit is the approximation of data using a linear function.

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Introduction, continued

The slope of a linear function is the change in the

dependent variable divided by the change in the

independent variable, or , sometimes written

as .

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Introduction, continued

The slope between two points (x1, y1) and (x2, y2) is

and the slope in the equation y = mx + b is m. The slope describes how

much y changes when x changes by 1. When analyzing the slope in the

context of a real-world situation, remember to use the units of x and y in the

calculation of the slope.

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Guided Practice

Example 1Andrew wants to estimate his gas mileage, or miles traveled per gallon of gas used. He records the number of gallons of gas he purchased and the total miles he traveled with that gas. (Create a

Scatter plot)

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Gallons Miles

15 313

17 340

18 401

19 423

18 392

17 379

20 408

19 437

16 366

20 416

Guided Practice: Example 1, continuedCreate a scatter plot showing the relationship between gallons of gas and miles driven. Which function is a better estimate for the function that relates gallons to miles: y = 10x or y = 22x? How is the equation of the function related to his gas mileage?

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Guided Practice: Example 1, continued1. Plot each point on the coordinate plane.

Let the x-axis represent gallons and the y-axis represent miles.

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Mil

es

Gallons

Guided Practice: Example 1, continued2. Graph the function y = 10x on the

coordinate plane. It is a linear function, so only two points are needed to draw the line. Evaluate the function at two values of x, such as 0 and 10, and draw a line through these points on the scatter plot.

y = 10xy = 10(0) = 0 Substitute 0 for x.y = 10(10) = 100 Substitute 10 for x.

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Guided Practice: Example 1, continuedTwo points on the line are (0, 0) and (10, 100).

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Mil

es

Gallons

Guided Practice: Example 1, continued3. Graph the function y = 22x on the same

coordinate plane.This is also a linear function, so only two points are needed to draw the line.Evaluate the function at two values of x, such as 0 and 10, and draw a line through these points on the scatter plot.

y = 22(0) = 0 Substitute 0 for x.y = 22(10) = 220 Substitute 10 for x.

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Guided Practice: Example 1, continuedTwo points on the line are (0, 0) and (0, 220).

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Mil

es

Gallons

Guided Practice: Example 1, continued4. Look at the graph of the data and the

functions. Identify which function comes closer to the data values. This function is the better estimate for the data. The graph of the function y = 22x goes through approximately the center of the points in the scatter plot. The function y = 10x is not steep enough to match the data values. The function y = 22x is a better estimate of the data.

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Guided Practice: Example 1, continued5. Interpret the equation in the context of

the problem, using the units of the x- and y-axes.For a linear equation in the form y = mx + b, the slope (m) of the equation is the rate of change of the function, or the change in y over the change in x. The y-intercept (b) of the equation is the initial value.In this example, y is miles and x is gallons. The slope is

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Guided Practice: Example 1, continued

For the equation y = 22x, the slope of 22 is equal to .

The gas mileage of Andrew’s car is the miles driven per gallon of gas used. The gas mileage is equal to the slope of the line that fits the data.

Andrew’s car has a gas mileage of approximately 22 miles per gallon.

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Guided Practice

Example 2The principal at Park High School records the total number of students each year. The table to the right shows the number of students for each of the last 8 years.

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Year Number of students

1 630

2 655

3 690

4 731

5 752

6 800

7 844

8 930

Guided Practice: Example 2, continuedCreate a scatter plot showing the relationship between the year and the total number of students. Show that the function y = 600(1.05)x is a good estimate for the relationship between the year and the population. Approximately how many students will attend the high school in year 9?

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1. Plot each point on the coordinate plane.

Let the x-axis represent years and the y-axis represent the number of students.

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Nu

mb

er

of

stu

de

nts

Year


2. Graph y = 600(1.05)x on the coordinate plane.

Calculate the value of y for a few different values of x. Start with x = 0. Calculate the value of the function for at least four more x-values from the data table.

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x y

0 600(1.05)0 = 600

1 600(1.05)1 = 630

3 600(1.05)3 = 694.575

5 600(1.05)5 = 765.769

7 600(1.05)7 = 855.260

Guided Practice: Example 2, continuedPlot these points on the same coordinate plane. Connect the points with a curve.

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Nu

mb

er

of

stu

de

nts

Year


3. Compare the graph of the function to the scatter plot of the data.

The graph of the function appears to be very close to the points in the scatter plot. The function y = 600(1.05)x is a good estimate of the data.

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Guided Practice: Example 2, continued4. Use the function to estimate the

population in year 9. Evaluate the function y = 600(1.05)x for year 9, when x = 9.

y = 600(1.05)9 = 930.797The function y = 600(1.05)x is a good estimate of the population. There will be approximately 931 students in the school in year 9.

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solving problems given functions fitted to data/interpreting slope and y-intercept key terms: linear...

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