Holt Algebra Modeling Real-World Data Warm Up quadratic: y ≈ 2.13x 2 – 2x x35813 y Use a calculator to perform quadratic and exponential regressions on the following data. exponential: y ≈ 10.57(1.32) x

Holt Algebra Modeling Real-World Data Apply functions to problem situations. Use mathematical models to make predictions. Objectives

Holt Algebra Modeling Real-World Data Use constant differences or ratios to determine which parent function would best model the given data set. Example 1A: Identifying Models by Using Constant Differences or Ratios Notice that the time data are evenly spaced. Check the first differences between the heights to see if the data set is linear. Time (yr) Height (in.)

Holt Algebra Modeling Real-World Data Example 1A Continued Because the first differences are a constant 35, a linear model will best model the data. Height (in.) First differences

Holt Algebra Modeling Real-World Data Example 1B: Identifying Models by Using Constant Differences or Ratios Time (yr) Population10,0009,6009,2168,8478,493 First differences –400 –384 –369 –354 Notice that the time data are evenly spaced. Check the first differences between the populations. Population10,0009,6009,2168,8478,493 Second differences Neither the first nor second differences are constant. SOOOOO…..Check ratios between the volumes.

Holt Algebra Modeling Real-World Data Example 1B Continued Neither the first nor second differences are constant. Check ratios between the volumes. Population10,0009,6009,2168,8478,493 Because the ratios between the values of the dependent variable are constant, an exponential function would best model the data.

Holt Algebra Modeling Real-World Data Example 1B Continued Check A scatter plot reveals a shape similar to an exponential decay function. Write an equation

Holt Algebra Modeling Real-World Data Example 1C: Identifying Models by Using Constant Differences or Ratios Time (s)12345 Height (m) First differences 33 –11 –55 –99 Notice that the time data are evenly spaced. Check the first differences between the heights. Height (m) Second differences –44 –44 –44

Holt Algebra Modeling Real-World Data Example 1C Continued Because the second differences of the dependent variables are constant when the independent variables are evenly spaced, a quadratic function will best model the data. Check A scatter reveals a shape similar to a quadratic parent function f(x) = x 2.

Holt Algebra Modeling Real-World Data Use constant differences or ratios to determine which parent function would best model the given data set. Notice that the data for the y-values are evenly spaced. Check the first differences between the x-values. x y Check It Out! Example 1a

Holt Algebra Modeling Real-World Data x First differences Check It Out! Example 1a Continued Second differences Because the second differences of the independent variable are constant when the dependent variables are evenly spaced, a square-root function will best model the data.

Holt Algebra Modeling Real-World Data x y First differences Notice that x-values are evenly spaced. Check the differences between the y-values. y Second differences Check It Out! Example 1b

Holt Algebra Modeling Real-World Data Neither the first nor second differences are constant. Check ratios between the values. Check It Out! Example 1b Continued

Holt Algebra Modeling Real-World Data Because the ratios between the values of the dependent variable are constant, an exponential function would best model the data. Check A scatter reveals a shape similar to an exponential decay function. Check It Out! Example 1b Continued

Holt Algebra Modeling Real-World Data To display the correlation coefficient r on some calculators, you must turn on the diagnostic mode. Press, and choose DiagnosticOn. Helpful Hint

Holt Algebra Modeling Real-World Data A printing company prints advertising flyers and tracks its profits. Write a function that models the given data. Example 2: Economic Application Flyers Printed Profit ($) Step 1 Make a scatter plot of the data. The data appear to form a quadratic or exponential pattern.

Holt Algebra Modeling Real-World Data Example 2 Continued Profit ($) Step 1 Analyze differences. First differences Second differences Because the second differences of the dependent variable are close to constant when the independent variables are evenly spaced, a quadratic function will best model the data.

Holt Algebra Modeling Real-World Data Example 2 Continued Step 3 Use your graphing calculator to perform a quadratic regression. A quadratic function that models the data is f(x) = 0.002x x – The correlation r is very close to 1, which indicates a good fit.

Holt Algebra Modeling Real-World Data Write a function that models the given data. x y Step 1 Make a scatter plot of the data. The data appear to form a quadratic or exponential pattern. Check It Out! Example 2

Holt Algebra Modeling Real-World Data Step 1 Analyze differences. First differences Second differences Because the second differences of the dependent variable are constant when the independent variables are evenly spaced, a quadratic function will best model the data. Check It Out! Example 2 Continued y

Holt Algebra Modeling Real-World Data Step 3 Use your graphing calculator to perform a quadratic regression. A quadratic function that models the data is f(x) = 0.5x x + 8. The correlation r is 1, which indicates a good fit. Check It Out! Example 2 Continued

Holt Algebra Modeling Real-World Data When data are not ordered or evenly spaced, you may to have to try several models to determine which best approximates the data. Graphing calculators often indicate the value of the coefficient of determination, indicated by r 2 or R 2. The closer the coefficient is to 1, the better the model approximates the data.

Holt Algebra Modeling Real-World Data The data shows the population of a small town since Using 1990 as a reference year, write a function that models the data. Example 3: Application The data are not evenly spaced, so you cannot analyze differences or ratios. Year Population

Holt Algebra Modeling Real-World Data Example 3 Continued Population year Create a scatter plot of the data. Use 1990 as year 0. The data appear to be quadratic, cubic or exponential.

Holt Algebra Modeling Real-World Data Use the calculator to perform each type of regression. Example 3 Continued Compare the value of r 2. The exponential model seems to be the best fit. The function f(x) ≈ (1.07) x models the data.

Holt Algebra Modeling Real-World Data Write a function that models the data. The data are not evenly spaced, so you cannot analyze differences or ratios. Fertilizer/Acre (lb) Yield/Acre (bushels) Check It Out! Example 3

Holt Algebra Modeling Real-World Data Check It Out! Example 3 Continued Create a scatter plot of the data. The data appear to be quadratic, cubic or exponential.

Holt Algebra Modeling Real-World Data Use the calculator to perform each type of regression. The function f(x) ≈ –9.3x 3 –0.2x x Check It Out! Example 3 Continued Compare the value of r 2 The cubic model seems to be the best fit. QuadraticExponentialCubic

Holt Algebra Modeling Real-World Data Lesson Quiz quadratic; C(l) = 2.5x Use finite differences or ratios to determine which parent function would best model the given data set, and write a function that models the data. Length (ft) Cost ($) f(x) = 1.085(1.977) x 2. Write a function that models the given data. Time (min) Volume (cm 2 )