Section 2 2 Finding Equations of Linear Models

Section 2. 2 Finding Equations of Linear Models

Finding an Equation of a Linear Model by Using Data Example The average number of credit card offers a household receives in one month is increased approximately linearly from 5. 1 offers in 2002 to 5. 9 offers in 2005 (Source: Synovate). Let n be the average number of credit card offers a household receives in one month at t years since 2000. Find the model. Solution The known values are shown (right). Section 2. 2 Lehmann, Intermediate Algebra, 3 ed Slide 2

Finding an Equation of a Linear Model by Using Data Solution Continued • Linear function can be put into the form • Here y depends on x • t and n are approximately linear • So, t depends on n • Thus, our model equation is • Now we find the slope: Section 2. 2 Lehmann, Intermediate Algebra, 3 ed Slide 3

Finding an Equation of a Linear Model by Using Data Solution Continued • Substitute 0. 27 for m in the equation : • Now we need to find b • Substitute one of the coordinates and solve for b. • Substituting into : Section 2. 2 Lehmann, Intermediate Algebra, 3 ed Slide 4

Finding an Equation of a Linear Model by Using Data Solution Continued • Substitute 4. 56 for b in the equation : Solution • Graphing Now we need to find • Verify using TRACE checking (2, 5. 1) and (5, 5. 9) Section 2. 2 Lehmann, Intermediate Algebra, 3 ed Slide 5

Finding an Equation of a Linear Model By Using Data Finding an Equation of a Linear Model by Using Data Example During the 1900 s there was great consumer demand for food products claiming to be “low fat” or “no fat. ” Since then, this demand has declined greatly. The table (on slide 7) shows the percentage of new food products claiming to be “low fat” or “no fat” from 1996 to 2001. Let p be the percentage of new food products claiming to be “low fat” or “no fat” at t years since 1995. Find an equation of a line that comes close to the points in the scattergram of data. Section 2. 2 Lehmann, Intermediate Algebra, 3 ed Slide 6

Finding an Equation of a Linear Model By Using Data Finding an Equation of a Linear Model by Using Data Solution • View point positions in the scattergram • Use a graphing calculator • Saves time and improves accuracy Section 2. 2 Lehmann, Intermediate Algebra, 3 ed Slide 7

Finding an Equation of a Linear Model By Using Data Finding an Equation of a Linear Model by Using Data Solution Continued • Red line contains points (4, 17) and (5, 16) does not come close to the other data points • Green line contains the points (1, 29) and (3, 22) and appears to come close to the rest of the points • So, we must find the equation of the green line Section 2. 2 Lehmann, Intermediate Algebra, 3 ed Slide 8

Finding an Equation of a Linear Model By Using Data Finding an Equation of a Linear Model by Using Data Solution Continued • Use the points (1, 29) and (3, 22) to find the slope: • Substitute – 3. 5 for m: Section 2. 2 Lehmann, Intermediate Algebra, 3 ed Slide 9

Finding an Equation of a Linear Model By Using Data Finding an Equation of a Linear Model by Using Data Solution Continued • To find b substitute the point (1, 29) into the equation and then solve for b. • Substituting 32. 5 for b: Section 2. 2 Lehmann, Intermediate Algebra, 3 ed Slide 10

Finding an Equation of a Linear Model By Using Data Finding an Equation of a Linear Model by Using Data Graphing Calculator • Check correctness of equation using graphing calculator • Verify that the line contains (1, 29) and (3, 22) Section 2. 2 Lehmann, Intermediate Algebra, 3 ed Slide 11

Finding an Equation of a Linear Line Finding an Equation of a Linear Model by Using Data Process To find an equation of a linear model, given some data: 1. Create a scattergram of the data. 2. Determine whethere is a line that comes close to the data points. If so, choose two points (not necessarily data points) that you can use to find the equation of a linear function. 3. Find an equation of the line you identified. Section 2. 2 Lehmann, Intermediate Algebra, 3 ed Slide 12

Finding an Equation of a Linear Line Finding an Equation of a Linear Model by Using Data Process Continued 4. Use a graphing calculator to verify that the graph of your equation comes close to the point of the scattergram. Graphing Calculator • Linear equation found by linear regression are called linear regression equations/functions • Most graphing calculators have regression features Section 2. 2 Lehmann, Intermediate Algebra, 3 ed Slide 13

Finding Linear Equations of a Linear Model Finding an Equation of a Linear Model by Using Data Example Cigarette smoking has been on the decline for the past several decades. Let p be the percentage of Americans who smoke at t years since 1900. 1. Use two well-chosen points to find an equation of a model that describes the relationship between t and p. 2. Find the linear regression equation and line by using a graphing calculator. Compare this model with the one your found in the part 1. Section 2. 2 Lehmann, Intermediate Algebra, 3 ed Slide 14

Finding Linear Equations of a Linear Model Finding an Equation of a Linear Model by Using Data Solution • Using the points (70, 37. 4) and (105, 19. 0) to calculate the slope: • Equation is of the form: Section 2. 2 Lehmann, Intermediate Algebra, 3 ed Slide 15

Finding Linear Equations of a Linear Model Finding an Equation of a Linear Model by Using Data Solution Continued • To find b, substitute the point (70, 37. 4) into the equation : • Equation is • Use graphing calculator to verify that the linear model contains the points (70, 37. 4) and (105, 19. 0) Section 2. 2 Lehmann, Intermediate Algebra, 3 ed Slide 16

Finding Linear Equations of a Linear Model Finding an Equation of a Linear Model by Using Data Solution Continued Comparing solution to the first example: is close to Section 2. 2 Lehmann, Intermediate Algebra, 3 ed Slide 17