holt algebra 2 9-6 modeling real-world data 9-6 modeling real-world data holt algebra 2 warm up warm...
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Holt Algebra 2
9-6 Modeling Real-World Data9-6 Modeling Real-World Data
Holt Algebra 2
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
Holt Algebra 2
9-6 Modeling Real-World Data
Warm Up
quadratic: y ≈ 2.13x2 – 2x + 6.12
x 3 5 8 13
y 19 50 126 340
1. Use a calculator to perform quadratic and exponential regressions on the following data.
exponential: y ≈ 10.57(1.32)x
Holt Algebra 2
9-6 Modeling Real-World Data
Apply functions to problem situations.
Use mathematical models to make predictions.
Objectives
Holt Algebra 2
9-6 Modeling Real-World Data
Much of the data that you encounter in the real world may form a pattern. Many times the pattern of the data can be modeled by one of the functions you have studied. You can then use the functions to analyze trends and make predictions. Recall some of the parent functions that you have studied so far.
Holt Algebra 2
9-6 Modeling Real-World Data
Holt Algebra 2
9-6 Modeling Real-World Data
Because the square-root function is the inverse of the quadratic function, the constant differences for x- and y-values are switched.
Helpful Hint
Holt Algebra 2
9-6 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) 5 10 15 20 25
Height (in.) 58 93 128 163 198
Holt Algebra 2
9-6 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.) 58 93 128 163 198
First differences 35 35 35 35 .
Holt Algebra 2
9-6 Modeling Real-World Data
Example 1B: Identifying Models by Using Constant Differences or Ratios
Time (yr) 4 8 12 16 20
Population 10,000 9,600 9,216 8,847 8,493
First differences –400 –384 –369 –354
Notice that the time data are evenly spaced. Check the first differences between the populations.
Population 10,000 9,600 9,216 8,847 8,493
Second differences 16 15 15
Holt Algebra 2
9-6 Modeling Real-World Data
Example 1B Continued
and ≈ 0.96. 8493 8847
Neither the first nor second differences are constant. Check ratios between the volumes.
9216 9600
8847 9216
= 0.96, 9600 10,000
= 0.96, ≈ 0.96,
Holt Algebra 2
9-6 Modeling Real-World Data
Example 1B Continued
Because the ratios between the values of the dependent variable are constant, an exponential function would best model the data.
Check A scatter plot reveals a shape similar to an exponential decay function.
Holt Algebra 2
9-6 Modeling Real-World Data
Example 1C: Identifying Models by Using Constant Differences or Ratios
Time (s) 1 2 3 4 5
Height (m) 132 165 154 99 0
First differences 33 –11 –55 –99
Notice that the time data are evenly spaced. Check the first differences between the heights.
Height (m) 132 165 154 99 0
Second differences –44 –44 –44
Holt Algebra 2
9-6 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) = x2.
Holt Algebra 2
9-6 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 12 48 108 192 300
y 10 20 30 40 50
Check It Out! Example 1a
Holt Algebra 2
9-6 Modeling Real-World Data
x 12 48 108 192 300
First differences 36 60 84 108
Check It Out! Example 1a Continued
Second differences 24 24 24
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 2
9-6 Modeling Real-World Data
x 21 22 23 24
y 243 324 432 576
First differences 81 108 144
Notice that x-values are evenly spaced. Check the differences between the y-values.
y 243 324 432 576
Second differences 27 36
Check It Out! Example 1b
Holt Algebra 2
9-6 Modeling Real-World Data
Neither the first nor second differences are constant. Check ratios between the values.
Check It Out! Example 1b Continued
324 243
≈ 1.33, and≈ 1.33, ≈ 1.33 432 324
576 432
Holt Algebra 2
9-6 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 2
9-6 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 2
9-6 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 100 200 300 400 500 600
Profit ($) 10 70 175 312 500 720
Step 1 Make a scatter plot of the data. The data appear to form a quadratic or exponential pattern.
Holt Algebra 2
9-6 Modeling Real-World Data
Example 2 Continued
Profit ($) 10 70 175 312 500 720
Step 1 Analyze differences.
First differences 60 105 137 188 220
Second differences 45 32 51 32
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 2
9-6 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) = 0.002x2 + 0.007x – 11.2. The correlation r is very close to 1, which indicates a good fit.
Holt Algebra 2
9-6 Modeling Real-World Data
Write a function that models the given data.
x 12 14 16 18 20 22 24
y 110 141 176 215 258 305 356
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 2
9-6 Modeling Real-World Data
Step 1 Analyze differences.
First differences 31 35 39 43 47 51
Second differences 4 4 4 4 4
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 110 141 176 215 258 305 356
Holt Algebra 2
9-6 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) = 0.5x2 + 2.5x + 8. The correlation r is 1, which indicates a good fit.
Check It Out! Example 2 Continued
Holt Algebra 2
9-6 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 r2 or R2. The closer the coefficient is to 1, the better the model approximates the data.
Holt Algebra 2
9-6 Modeling Real-World Data
The data shows the population of a small town since 1990. 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 1990 1993 1997 2000 2002 2005 2006
Population 400 490 642 787 901 1104 1181
Holt Algebra 2
9-6 Modeling Real-World Data
Create a scatter plot of the data. Use 1990 as year 0. The data appear to be quadratic, cubic or exponential.
Example 3 Continued
Pop
ulat
ion
year1990
20061997
20022005
1993
400
800
1200
2000
••
•
••
• •
Holt Algebra 2
9-6 Modeling Real-World Data
Use the calculator to perform each type of regression.
Example 3 Continued
Compare the value of r2. The exponential model seems to be the best fit. The function f(x) ≈ 399.94(1.07)x models the data.
Holt Algebra 2
9-6 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) 11 14 25 31 40 50
Yield/Acre (bushels) 245 302 480 557 645 705
Check It Out! Example 3
Holt Algebra 2
9-6 Modeling Real-World Data
Create a scatter plot of the data. The data appear to be quadratic, cubic or exponential.
Check It Out! Example 3 Continued
Holt Algebra 2
9-6 Modeling Real-World Data
Use the calculator to perform each type of regression.
Compare the value of r2. The cubic model seems to be the best fit. The function f(x) ≈ –9.3x3 –0.2x2 + 23.97x + 5.46.
Check It Out! Example 3 Continued
Holt Algebra 2
9-6 Modeling Real-World Data
Lesson Quiz
quadratic; C(l) = 2.5x2 + 40
1. 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) 5 10 15 20 25
Cost ($) 102.50 290.00 602.50 1040.00 1602.50
f(x) = 1.085(1.977)x
2. Write a function that models the given data.
Time (min) 0 2 4 6 8 10
Volume (cm2) 1.2 3.9 15.7 64.2 256.5 1023.8