holt algebra 2 7-8 curve fitting with exponential and logarithmic models 7-8 curve fitting with...
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Holt Algebra 2
7-8 Curve Fitting with Exponentialand Logarithmic Models7-8 Curve Fitting with Exponential and Logarithmic Models
Holt Algebra 2
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
Holt Algebra 2
7-8 Curve Fitting with Exponentialand Logarithmic Models
Warm UpPerform a quadratic regression on the following data:
x 1 2 6 11 13
f(x) 3 6 39 120 170
f(x) ≈ 0.98x2 + 0.1x + 2.1
Holt Algebra 2
7-8 Curve Fitting with Exponentialand Logarithmic Models
Model data by using exponential and logarithmic functions.
Use exponential and logarithmic models to analyze and predict.
Objectives
Holt Algebra 2
7-8 Curve Fitting with Exponentialand Logarithmic Models
exponential regressionlogarithmic regression
Vocabulary
Holt Algebra 2
7-8 Curve Fitting with Exponentialand Logarithmic Models
Analyzing data values can identify a pattern, or repeated relationship, between two quantities.
Look at this table of values for the exponential function f(x) = 2(3x).
Holt Algebra 2
7-8 Curve Fitting with Exponentialand Logarithmic Models
For linear functions (first degree), first differences are constant. For quadratic functions, second differences are constant, and so on.
Remember!
Notice that the ratio of each y-value and the previous one is constant. Each value is three times the one before it, so the ratio of function values is constant for equally spaced x-values. This data can be fit by an exponential function of the form f(x) = abx.
Holt Algebra 2
7-8 Curve Fitting with Exponentialand Logarithmic Models
Determine whether f is an exponential function of x of the form f(x) = abx. If so, find the constant ratio.
Example 1: Identifying Exponential Data
A.
+1 +2 +3 +4
x –1 0 1 2 3
f(x) 2 3 5 8 12
B. x –1 0 1 2 3
f(x) 16 24 36 54 81
+8 +12 +18 +27FirstDifferences
SecondDifferences
+1 +1 +1 +4 +6 +9
Ratio 81 54
=54 36
=24 16
=36 24
= 3 2
Second differences are constant; f is a quadratic function of x.
This data set is exponential, with a constant ratio of 1.5.
Holt Algebra 2
7-8 Curve Fitting with Exponentialand Logarithmic Models
Determine whether y is an exponential function of x of the form f(x) = abx. If so, find the constant ratio.a.
x –1 0 1 2 3
f(x) 2.6 4 6 9 13.5
b.
x –1 0 1 2 3
f(x) –3 2 7 12 17
+5 +5 +5 +5FirstDifferences
SecondDifferences
+0.66 +1 +1.5
Ratio13.5 9 =
9 6 =
6 4 =
2 1.3
First differences are constant; y is a linear function of x.
This data set is exponential, with a constant ratio of 1.5.
Check It Out! Example 1
+1.34 +2 +3 +4.5
4 2.6
=
Holt Algebra 2
7-8 Curve Fitting with Exponentialand Logarithmic Models
In Chapters 2 and 5, you used a graphing calculator to perform linear progressions and quadratic regressions to make predictions. You can also use an exponential model, which is an exponential function that represents a real data set.
Once you know that data are exponential, you can use ExpReg (exponential regression) on your calculator to find a function that fits. This method of using data to find an exponential model is called an exponential regression. The calculator fits exponential functions to abx, so translations cannot be modeled.
Holt Algebra 2
7-8 Curve Fitting with Exponentialand Logarithmic Models
If you do not see r2 and r when you calculate regression, and turn these on by selecting DiagnosticOn.
Remember!
Holt Algebra 2
7-8 Curve Fitting with Exponentialand Logarithmic Models
Find an exponential model for the data. Use the model to predict when the tuition at U.T. Austin will be $6000.
Example 2: College Application
Step 1 Enter data into two lists in a graphing calculator. Use the exponential regression feature.
Tuition of the University of Texas
Year Tuition
1999–00 $3128
2000–01 $3585
2001–02 $3776
2002–03 $3950
2003–04 $4188
An exponential model is f(x) ≈ 3236(1.07t), where f(x) represents the tuition and t is the number of years after the 1999–2000 year.
Holt Algebra 2
7-8 Curve Fitting with Exponentialand Logarithmic Models
Example 2 Continued
Step 2 Graph the data and the function model to verify that it fits the data.
To enter the regression equation as Y1 from the screen, press , choose 5:Statistics, press , scroll to select the EQ menu, and select 1:RegEQ.
Holt Algebra 2
7-8 Curve Fitting with Exponentialand Logarithmic Models
Enter 6000 as Y2. Use the intersection feature. You may need to adjust the dimensions to find the intersection.
The tuition will be about $6000 when t = 9 or 2008–09.
Example 2 Continued
7500
150
0
Holt Algebra 2
7-8 Curve Fitting with Exponentialand Logarithmic Models
Use exponential regression to find a function that models this data. When will the number of bacteria reach 2000?
Step 1 Enter data into two lists in a graphing calculator. Use the exponential regression feature.An exponential model is f(x) ≈ 199(1.25t), where f(x) represents the tuition and t is the number of minutes.
Check It Out! Example 2
Time (min) 0 1 2 3 4 5
Bacteria 200 248 312 390 489 610
Holt Algebra 2
7-8 Curve Fitting with Exponentialand Logarithmic Models
Step 2 Graph the data and the function model to verify that it fits the data.
Check It Out! Example 2 Continued
To enter the regression equation as Y1 from the screen, press , choose 5:Statistics, press , scroll to select the EQ menu, and select 1:RegEQ.
Holt Algebra 2
7-8 Curve Fitting with Exponentialand Logarithmic Models
Enter 2000 as Y2. Use the intersection feature. You may need to adjust the dimensions to find the intersection.
The bacteria count at 2000 will happen at approximately 10.3 minutes.
2500
00 15
Check It Out! Example 2 Continued
Holt Algebra 2
7-8 Curve Fitting with Exponentialand Logarithmic Models
Many natural phenomena can be modeled by natural log functions. You can use a logarithmic regression to find a function
Most calculators that perform logarithmic regression use ln rather than log.
Helpful Hint
Holt Algebra 2
7-8 Curve Fitting with Exponentialand Logarithmic Models
Find a natural log model for the data. According to the model, when will the global population exceed 9,000,000,000?
Enter the into the two lists in a graphing calculator. Then use the logarithmic regression feature. Press CALC 9:LnReg. A logarithmic model is f(x) ≈ 1824 + 106ln x, where f is the year and x is the population in billions.
Global Population Growth
Population (billions)
Year
1 1800
2 1927
3 1960
4 1974
5 1987
6 1999
Example 3: Application
Holt Algebra 2
7-8 Curve Fitting with Exponentialand Logarithmic Models
The calculated value of r2 shows that an equation fits the data.
Example 3 Continued
Graph the data and function model to verify that it fits the data.
Use the value feature to find y when x is 9. The population will exceed 9,000,000,000 in the year 2058.
0
2500
0 15
Holt Algebra 2
7-8 Curve Fitting with Exponentialand Logarithmic Models
Time (min) 1 2 3 4 5 6 7
Speed (m/s) 0.5 2.5 3.5 4.3 4.9 5.3 5.6
Use logarithmic regression to find a function that models this data. When will the speed reach 8.0 m/s?
Check It Out! Example 4
Enter the into the two lists in a graphing calculator. Then use the logarithmic regression feature. Press CALC 9: LnReg. A logarithmic model is f(x) ≈ 0.59 + 2.64 ln x, where f is the time and x is the speed.
Holt Algebra 2
7-8 Curve Fitting with Exponentialand Logarithmic Models
Check It Out! Example 4 Continued
The calculated value of r2 shows that an equation fits the data.
Graph the data and function model to verify that it fits the data. an equation fits the data.
Use the intersect feature to find y when x is 8. The time it will reach 8.0 m/s is 16.6 min.
0
10
200
Holt Algebra 2
7-8 Curve Fitting with Exponentialand Logarithmic Models
Lesson Quiz: Part I
Determine whether f is an exponential function of x. If so, find the constant ratio.
x –1 0 1 2
f(x) 10 9 8.1 7.291.
x –1 0 1 2 3
f(x) 3 6 12 21 33
yes; constant ratio = 0.9
no; second difference are constant; f is quadratic.
2.
Holt Algebra 2
7-8 Curve Fitting with Exponentialand Logarithmic Models
Lesson Quiz: Part II3. Find an exponential model for the data. Use the model to estimate when the insurance value will drop below $2000.
Insurance Value
Year(1990 = year 0) Value
0 10,000
2 9,032
5 7,753
9 6,290
11 5,685
f(x) ≈ 10,009(0.95)t; value will dip below 2000 in year 32 or 2022.