Exponential Growth and Decay. Section 6.3. Warm-up 1. -8 -6 -4 -2 0 2 4 6 8. As the x-values increase by 1, the y-values increase by 2. Warm-up 2. 7 0 -5 -8 -9 -8 -5 0 7. When x<1, as the x-values increase, the y-values decrease. When x>1, as the x-values increase, the
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Exponential Growth and Decay
As the x-values increase by 1, the y-values increase by 2.
When x<1, as the x-values increase, the
When x>1, as the x-values increase, the
Follow up Questions
(With your partner be prepared to answer the following questions about this activity)
Do the graphs represent a constant rate of change?
Do the graphs appear quadratic?
How do the tables verify your observations?
What is the y-intercept for each graph?
Where do you find the y-intercept recorded on the table? (in each problem)
What is the ratio between any two successive given y-values? (in each problem)
Is the successive quotient for each pair of y-values constant?(in each problem)
The data given in the table below is a result of placing a mat on a flat surface and dropping beans onto the mat.
For each bean that landed in a shaded area of the mat, you added one additional bean to your total number of beans.
Bean Drop Data
Create a scatterplot of the data and record your window. Use the Sky Is Falling tables you have been given to write a linear and an exponential model for your data. Round answers to the nearest thousandths.
Bean Drop Data
Window for this graph:
x-min: 0 x-max: 10 x-scale: 1
y-min: 0 y-max: 125 y-scale: 10
Attach the Sky is Falling Table into your notes
On your graphing calculator, graph both the linear and the exponential models you created.
Was there a constant rate of change for your models?
Was there a constant successive quotient for your models?
How did you determine the initial values from your tables?
Which one of the above models is a better fit for the data? Why?
Attach the Sky is Falling Table to your notes
y = 16.143x+10
y = 10(1.433)x
What is occurring mathematically in order for a situation to be modeled with a linear function?
What values are necessary to write a linear model?
What is occurring mathematically in order for a situation to be modeled with a exponential function?
What values are necessary to write an exponential model?
What is occurring mathematically in order for a situation to be modeled with neither a linear nor exponential function?
Constant Rate Of Change
Slope and y-intercept
Constant successive quotients
The “a” Initial Value and successive quotients to find the base “b”
Will not have a constant rate of change or constant successive quotients
How do we handle problems like 2 and 3 when given a rate, r%, of the growth or decay?
Are the dependent values in a situation increasing or decreasing each time?
How might this effect the base value in our exponential model?
Increasing values would show growth
Decreasing values would show decay
When a real-life quantity increases by a fixed percent each year (or other time period), the amount y of the quantity after t years can be modeled by the equation:
where a is the initial amount and r is the percent increase expressed as a decimal.
NOTE: b is the growth factor.
Sometimes the equation is written, A=P(1+r)t, where A stands for the balance amount and P stands for the principal, or initial amount.
When a real-life quantity decreases by a fixed percent each year (or other time period), the amount y of the quantity after t years can be modeled by the equation:
where a is the initial amount and r is the percent decrease expressed as a decimal.
NOTE: b is the decay factor.
Sometimes the equation is written, A=P(1 - r)t, where A stands for the balance amount and P stands for the principal, or initial amount.
a = 2573 r = .92
Growth, so b = (1+r)
t = 2003 –1996 = 7 years
Example 1: (continued)
In 1996, there were 2573 computer viruses and other computer security incidents. During the next 7 years, the number of incidents increased by about 92% each year.
d. Estimate the year when there were about 125,000 computer security incidents.
Using your calculator: t is approximately 6, so in the year 2002,
there were about 125,000 security incidents.
a = 4200 r = 0.1
Decay, so b = (1 – r)
Example 2: (continued)
A new snowmobile costs $4200. The value of the snowmobile decreases by 10% each year.
d. Estimate when the value of the snowmobile will be $2500.
Find the point of intersection.
After about 5 years
Absent Students – Notes 6.3
Attach this note page into your notebook
Complete all examples.