Holt Algebra 2 71 Exponential Functions, Growth, and Decay exponential function baseasymptote...
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Holt Algebra Exponential Functions, Growth, and Decay Growth that doubles every year can be modeled by using a function with a variable as an exponent. This function is known as an exponential function. The parent exponential function is f(x) = b x, where the base b is a constant and the exponent x is the independent variable.
Transcript of Holt Algebra 2 71 Exponential Functions, Growth, and Decay exponential function baseasymptote...
Slide 1exponential function
base asymptote
model growth and decay situations.
Objective
Notes
In 2000, the world population was 6.08 billion and was increasing at a rate 1.21% each year.
1. Write a function for world population. Does the function represent growth or decay?
2. Predict the population in 2020.
The value of a $3000 computer decreases about 30% each year.
3. Write a function for the computer’s value. Does the function represent growth or decay?
4. Predict the value in 4 years.
*
Exponential Functions, Growth, and Decay
*
Exponential Functions, Growth, and Decay
*
Exponential Functions, Growth, and Decay
*
Exponential Functions, Growth, and Decay
*
Negative exponents indicate a reciprocal. For example:
Remember!
In the function y = bx, y is a function of x because the value of y depends on the value of x.
Remember!
Tell whether the function shows growth or decay. Then graph.
Example 1A: Graphing Exponential Functions
Step 1 Find the value of the base.
*
Example 1A Continued
Step 2 Graph the function by using a table of values.
x
0
2
4
6
8
10
12
f(x)
10
5.6
3.2
1.8
1.0
0.6
0.3
Tell whether the function shows growth or decay. Then graph.
Example 1B: Graphing Exponential Functions
Step 1 Find the value of the base.
g(x) = 100(1.05)x
The base, 1.05, is greater than 1. This is an exponential growth function.
g(x) = 100(1.05)x
Exponential Functions, Growth, and Decay
Tell whether the function p(x) = 5(1.2x) shows growth or decay. Then graph.
Step 1 Find the value of the base.
Example 1C
p(x) = 5(1.2 x)
*
Exponential Functions, Growth, and Decay
Step 2 Graph the function by using a table of values.
Example 1C Continued
Exponential Functions, Growth, and Decay
*
Exponential Functions, Growth, and Decay
Clara invests $5000 in an account that pays 6.25% interest per year. What will her investment be worth in seven years?
Example 2A: Economics Application
Step 1 Write a function to model the growth in value of her investment.
f(t) = a(1 + r)t
Exponential growth function.
f(t) = 5000(1 + 0.0625)t
Exponential Functions, Growth, and Decay
Clara invests $5000 in an account that pays 6.25% interest per year. What will her investment be worth in seven years?
Example 2A: Economics Application
f(t) = 5000(1.0625)t
Exponential Functions, Growth, and Decay
In 1981, the Australian humpback whale population was 350 and increased at a rate of 14% each year since then. Write a function to model population growth.
P(t) = a(1 + r)t
P(t) = 350(1 + 0.14)t
Exponential Functions, Growth, and Decay
A city population, which was initially 15,500, has been dropping 3% a year. Write an exponential function. What will the population be after ten years?
Example 3: Depreciation Application
f(t) = 15,500(1 – 0.03)t
*
Notes
In 2000, the world population was 6.08 billion and was increasing at a rate 1.21% each year.
1. Write a function for world population. Does the function represent growth or decay?
P(t) = 6.08(1.0121)t
2. Use a graph to predict the population in 2020.
≈ 7.73 billion
The value of a $3000 computer decreases about 30% each year.
3. Write a function for the computer’s value. Does the function represent growth or decay?
4. Use a graph to predict the value in 4 years.
V(t)≈ 3000(0.7)t
Exponential Functions, Growth, and Decay
5. Tell whether the function shows growth or decay. Then graph.
Notes (continued)
f(x) = 8(½)x
base asymptote
model growth and decay situations.
Objective
Notes
In 2000, the world population was 6.08 billion and was increasing at a rate 1.21% each year.
1. Write a function for world population. Does the function represent growth or decay?
2. Predict the population in 2020.
The value of a $3000 computer decreases about 30% each year.
3. Write a function for the computer’s value. Does the function represent growth or decay?
4. Predict the value in 4 years.
*
Exponential Functions, Growth, and Decay
*
Exponential Functions, Growth, and Decay
*
Exponential Functions, Growth, and Decay
*
Exponential Functions, Growth, and Decay
*
Negative exponents indicate a reciprocal. For example:
Remember!
In the function y = bx, y is a function of x because the value of y depends on the value of x.
Remember!
Tell whether the function shows growth or decay. Then graph.
Example 1A: Graphing Exponential Functions
Step 1 Find the value of the base.
*
Example 1A Continued
Step 2 Graph the function by using a table of values.
x
0
2
4
6
8
10
12
f(x)
10
5.6
3.2
1.8
1.0
0.6
0.3
Tell whether the function shows growth or decay. Then graph.
Example 1B: Graphing Exponential Functions
Step 1 Find the value of the base.
g(x) = 100(1.05)x
The base, 1.05, is greater than 1. This is an exponential growth function.
g(x) = 100(1.05)x
Exponential Functions, Growth, and Decay
Tell whether the function p(x) = 5(1.2x) shows growth or decay. Then graph.
Step 1 Find the value of the base.
Example 1C
p(x) = 5(1.2 x)
*
Exponential Functions, Growth, and Decay
Step 2 Graph the function by using a table of values.
Example 1C Continued
Exponential Functions, Growth, and Decay
*
Exponential Functions, Growth, and Decay
Clara invests $5000 in an account that pays 6.25% interest per year. What will her investment be worth in seven years?
Example 2A: Economics Application
Step 1 Write a function to model the growth in value of her investment.
f(t) = a(1 + r)t
Exponential growth function.
f(t) = 5000(1 + 0.0625)t
Exponential Functions, Growth, and Decay
Clara invests $5000 in an account that pays 6.25% interest per year. What will her investment be worth in seven years?
Example 2A: Economics Application
f(t) = 5000(1.0625)t
Exponential Functions, Growth, and Decay
In 1981, the Australian humpback whale population was 350 and increased at a rate of 14% each year since then. Write a function to model population growth.
P(t) = a(1 + r)t
P(t) = 350(1 + 0.14)t
Exponential Functions, Growth, and Decay
A city population, which was initially 15,500, has been dropping 3% a year. Write an exponential function. What will the population be after ten years?
Example 3: Depreciation Application
f(t) = 15,500(1 – 0.03)t
*
Notes
In 2000, the world population was 6.08 billion and was increasing at a rate 1.21% each year.
1. Write a function for world population. Does the function represent growth or decay?
P(t) = 6.08(1.0121)t
2. Use a graph to predict the population in 2020.
≈ 7.73 billion
The value of a $3000 computer decreases about 30% each year.
3. Write a function for the computer’s value. Does the function represent growth or decay?
4. Use a graph to predict the value in 4 years.
V(t)≈ 3000(0.7)t
Exponential Functions, Growth, and Decay
5. Tell whether the function shows growth or decay. Then graph.
Notes (continued)
f(x) = 8(½)x