Exponential Growth vs. Decay: Identifying Base b
Grade 9 students in California Reveal Math Algebra 1 learn to identify exponential growth versus decay by examining the base b in f(x)=ab^x with a>0. When b>1, the output multiplies by b for each unit increase in x — this is growth. When 0<b<1, the output shrinks by factor b each step — this is decay. For example, f(x)=3(2)^x is growth since b=2>1; g(x)=5(1/3)^x is decay since b=1/3<1; and h(x)=100(1.08)^x is growth since b=1.08>1. Checking whether b is greater or less than 1 is the fastest classification method.
Key Concepts
For an exponential function $f(x) = ab^x$ with $a 0$, the base $b$ determines whether the function represents growth or decay :.
If $b 1$, the function models exponential growth (output increases as $x$ increases). If $0 < b < 1$, the function models exponential decay (output decreases as $x$ increases).
Common Questions
How do you determine if f(x)=ab^x represents growth or decay?
Look at the base b. If b>1, each output is larger than the previous, so the function represents growth. If 0<b<1, each output is smaller, so the function represents decay.
Is f(x)=3(2)^x exponential growth or decay?
Exponential growth, because b=2>1. The output multiplies by 2 for each unit increase in x.
Is g(x)=5(1/3)^x exponential growth or decay?
Exponential decay, because b=1/3 and 0<1/3<1. The output shrinks by a factor of 1/3 for each unit increase in x.
Is h(x)=100(1.08)^x growth or decay?
Exponential growth, because b=1.08>1. The output increases by a factor of 1.08 each step.
What does the base b represent in f(x)=ab^x?
The base b is the constant ratio by which the output is multiplied for every one-unit increase in x. If b>1, outputs grow; if 0<b<1, outputs shrink.
Which unit covers identifying the base b for growth and decay?
This skill is from Unit 8: Exponential Functions in California Reveal Math Algebra 1, Grade 9.