Question

exponential functions
graphic organizer
equations
$y = a(b)^x$
$y = 4(\frac{2}{3})^x$
$a = $
$b = $
growth or decay?
$y = 0.5(\frac{5}{2})^x$
$a = $
$b = $
growth or decay?
graphs
$a = $ y - intercept
$b = $ growth or decay rate $\frac{y_2}{y_1}$
graph of an exponential growth curve
$a = $
$b = $
equation:
graph of an exponential decay curve
$a = $
$b = $
equation:
tables
$a = $ y - intercept (y when $x = 0$)
$b = $ common ratio $\frac{y_2}{y_1}$
$a = $
$b = $
equation:

x	y
0	12
1	3
2	$\frac{3}{4}$
3	$\frac{3}{16}$

$a = $
$b = $
equation:

x	y
0	6
1	30
2	150
3	750

word problems
$a = $ initial value
$b = $ growth or decay rate (at what rate is the situation increasing or decreasing by?)
match the following word problems to their correct equation on the right.
______ patricia’s car has a starting value of $20,000 and depreciates at a rate of 33% each year.
______ there were 20,000 cases of the flu reported in january. it is expected that each month there will be a third of the cases as the previous month.
______ jenny made a $20,000 investment that is doing incredibly well. it is expected to triple each year.
______ ronny bought a highly sought - after vintage car for $20,000. the value of the car increases at a rate of 33% each year.
a. $y = 20,000(3)^x$
b. $y = 20,000(\frac{1}{3})^x$
c. $y = 20,000(1.33)^x$
d. $y = 20,000(0.67)^x$

Explanation:

Step1: Identify a, b for $y=4(\frac{2}{3})^x$

Compare to $y=a(b)^x$:
$a=4$, $b=\frac{2}{3}$. Since $0

Step2: Identify a, b for $y=0.5(\frac{5}{2})^x$

Compare to $y=a(b)^x$:
$a=0.5$, $b=\frac{5}{2}$. Since $b>1$, it is growth.

Step3: Analyze left growth graph

y-intercept (a) = 1. $b=\frac{\text{next } y}{\text{previous } y}=2$. Equation: $y=1(2)^x=2^x$.

Step4: Analyze right decay graph

y-intercept (a) = 4. $b=\frac{\text{next } y}{\text{previous } y}=\frac{1}{2}$. Equation: $y=4(\frac{1}{2})^x$.

Step5: Analyze left table

a = y when x=0: $a=12$. $b=\frac{3}{12}=\frac{1}{4}$. Equation: $y=12(\frac{1}{4})^x$.

Step6: Analyze right table

a = y when x=0: $a=6$. $b=\frac{30}{6}=5$. Equation: $y=6(5)^x$.

Step7: Match word problems

1. Patricia's car: Depreciates 33%, so $b=1-0.33=0.67$. Match D.
2. Flu cases: 1/3 of previous, $b=\frac{1}{3}$. Match B.
3. Jenny's investment: Triples yearly, $b=3$. Match A.
4. Ronny's car: Grows 33%, so $b=1+0.33=1.33$. Match C.

Answer:

EQUATIONS

Left ($y=4(\frac{2}{3})^x$)

$a=4$
$b=\frac{2}{3}$
Growth or Decay? Decay

Right ($y=0.5(\frac{5}{2})^x$)

$a=0.5$
$b=\frac{5}{2}$
Growth or Decay? Growth

GRAPHS

Left (growth graph)

$a=1$
$b=2$
Equation: $y=2^x$

Right (decay graph)

$a=4$
$b=\frac{1}{2}$
Equation: $y=4(\frac{1}{2})^x$

TABLES

Left table

$a=12$
$b=\frac{1}{4}$
Equation: $y=12(\frac{1}{4})^x$

Right table

$a=6$
$b=5$
Equation: $y=6(5)^x$

WORD PROBLEMS

Patricia's car: D. $y = 20,000(0.67)^x$
Flu cases: B. $y = 20,000(\frac{1}{3})^x$
Jenny's investment: A. $y = 20,000(3)^x$
Ronny's vintage car: C. $y = 20,000(1.33)^x$