7.2E: Exercises - Graphs and Properties of Exponential Growth and Decay Functions

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PROBLEM SET: GRAPHS AND PROPERTIES OF EXPONENTIAL GROWTH AND DECAY FUNCTIONS

In questions 1-4, let $t$ = time in years and $y$ = the value at time $t$ or $y$ = the size of the population at time $t$. The domain is the set of non-negative values for $t$; $t ≥ 0$, because $y$ represents a physical quantity and negative values for time may not make sense. For each question:

Write the formula for the function in the form $y = ab^t$
Sketch the graph of the function and mark the coordinates of the y-intercept.

A house was purchased for $350,000 in the year 2010. The value has been increasing at the rate of 2% per year.	A population of a certain species of bird in a state park has 300 birds. The population is decreasing at the rate of 7% year.
A lab buys equipment $50,000. Its value depreciates over time. The value decreases at the rate of 6% annually.	A population of bats in a cave has 200 bats. The population is increasing at the rate of 5% annually.

In questions 5-8, let $t$ = time in years and $y$ = the value at time $t$ or $y$ = the size of the population at time $t$. The domain is the set of non-negative values for $t$; $t ≥ 0$, because $y$ represents a physical quantity and negative values for time may not make sense. For each question:

Write the formula for the function in the form $y = ae^{kt}$
Sketch the graph of the function and mark the coordinates of the y-intercept.

A population of 400 microbes increases at the continuous growth rate of 26% per day.	The price of a machine needed by a production factory is $28,000. Due to inflation the price of the machine is increasing at the continuous rate of 3.5% per year.
A population of an endangered species consists of 4000 animals of that species. The population is decreasing at the continuous rate of 12% per year.	A business buys a computer system for $12000. The value of the system is depreciating and decreases at the continuous rate of 20% per year.

For questions 9-12

Sketch a graph of exponential function.
List the coordinates of the y intercept.
State the equation of any asymptotes and state the whether the function approaches the asymptote as x →∞ or as x→ −∞ .
State the domain and range.

$y=10\left(1.5^{x}\right)$	$y=10\left(e^{1.2 x}\right)$
$y=32\left(0.75^{x}\right)$	$y=200\left(e^{-0.5 x}\right)$

\(y=10\left(1.5^{x}\right)\)	\(y=10\left(e^{1.2 x}\right)\)
\(y=32\left(0.75^{x}\right)\)	\(y=200\left(e^{-0.5 x}\right)\)

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