1. Solving Problems Given Functions Fitted to Data/Interpreting Slope and y-intercept Key Terms: Linear Fit Slope Scatter Plot 1 4.6: Solving Problems Given Functions Fitted to Data

2. In this lesson we will be able to… 1.Evaluate or solve a function via graph or x,y table 2. 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 2 4.6: Solving Problems Given Functions Fitted to Data

3. 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. 3 4.6: Solving Problems Given Functions Fitted to Data

4. Introduction When 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. 4 4.6: Solving Problems Given Functions Fitted to Data

5. 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. 5 4.6: Solving Problems Given Functions Fitted to Data

6. Introduction, continued The slope between two points (x 1, y 1 ) and (x 2, y 2 ) 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. 6 4.6: Solving Problems Given Functions Fitted to Data

7. Guided Practice Example 1 Andrew 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) 7 4.6: Solving Problems Given Functions Fitted to Data GallonsMiles 15313 17340 18401 19423 18392 17379 20408 19437 16366 20416

8. Guided Practice: Example 1, continued Create 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? 8 4.6: Solving Problems Given Functions Fitted to Data

9. Guided Practice: Example 1, continued 1.Plot each point on the coordinate plane. Let the x-axis represent gallons and the y-axis represent miles. 9 4.6: Solving Problems Given Functions Fitted to Data Miles Gallons

10. Guided Practice: Example 1, continued 2.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 = 10x y = 10(0) = 0Substitute 0 for x. y = 10(10) = 100 Substitute 10 for x. 10 4.6: Solving Problems Given Functions Fitted to Data

11. Guided Practice: Example 1, continued Two points on the line are (0, 0) and (10, 100). 11 4.6: Solving Problems Given Functions Fitted to Data Miles Gallons

12. Guided Practice: Example 1, continued 3.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. 12 4.6: Solving Problems Given Functions Fitted to Data

13. Guided Practice: Example 1, continued Two points on the line are (0, 0) and (0, 220). 13 4.6: Solving Problems Given Functions Fitted to Data Miles Gallons

14. Guided Practice: Example 1, continued 4.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. 14 4.6: Solving Problems Given Functions Fitted to Data

15. Guided Practice: Example 1, continued 5.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 15 4.6: Solving Problems Given Functions Fitted to Data

16. 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. 16 4.6: Solving Problems Given Functions Fitted to Data ✔

17. Guided Practice Example 2 The 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. 17 4.6: Solving Problems Given Functions Fitted to Data YearNumber of students 1630 2655 3690 4731 5752 6800 7844 8930

18. Guided Practice: Example 2, continued Create 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? 18 4.6: Solving Problems Given Functions Fitted to Data

19. Guided Practice: Example 2, continued 1.Plot each point on the coordinate plane. Let the x-axis represent years and the y-axis represent the number of students. 19 4.6: Solving Problems Given Functions Fitted to Data Number of students Year

20. Guided Practice: Example 2, continued 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. 20 4.6: Solving Problems Given Functions Fitted to Data xy 0600(1.05) 0 = 600 1600(1.05) 1 = 630 3600(1.05) 3 = 694.575 5600(1.05) 5 = 765.769 7600(1.05) 7 = 855.260

21. Guided Practice: Example 2, continued Plot these points on the same coordinate plane. Connect the points with a curve. 21 4.6: Solving Problems Given Functions Fitted to Data Number of students Year

22. Guided Practice: Example 2, continued 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. 22 4.6: Solving Problems Given Functions Fitted to Data

23. Guided Practice: Example 2, continued 4.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 = 930.797 The 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. 23 4.6: Solving Problems Given Functions Fitted to Data ✔