# 6.1 Exponential functions  (Page 10/16)

 Page 10 / 16

## Algebraic

For the following exercises, identify whether the statement represents an exponential function. Explain.

The average annual population increase of a pack of wolves is 25.

A population of bacteria decreases by a factor of $\text{\hspace{0.17em}}\frac{1}{8}\text{\hspace{0.17em}}$ every $\text{\hspace{0.17em}}24\text{\hspace{0.17em}}$ hours.

exponential; the population decreases by a proportional rate. .

The value of a coin collection has increased by $\text{\hspace{0.17em}}3.25%\text{\hspace{0.17em}}$ annually over the last $\text{\hspace{0.17em}}20\text{\hspace{0.17em}}$ years.

For each training session, a personal trainer charges his clients $\text{\hspace{0.17em}}\text{}5\text{\hspace{0.17em}}$ less than the previous training session.

not exponential; the charge decreases by a constant amount each visit, so the statement represents a linear function. .

The height of a projectile at time $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is represented by the function $\text{\hspace{0.17em}}h\left(t\right)=-4.9{t}^{2}+18t+40.$

For the following exercises, consider this scenario: For each year $\text{\hspace{0.17em}}t,$ the population of a forest of trees is represented by the function $\text{\hspace{0.17em}}A\left(t\right)=115{\left(1.025\right)}^{t}.\text{\hspace{0.17em}}$ In a neighboring forest, the population of the same type of tree is represented by the function $\text{\hspace{0.17em}}B\left(t\right)=82{\left(1.029\right)}^{t}.\text{\hspace{0.17em}}$ (Round answers to the nearest whole number.)

Which forest’s population is growing at a faster rate?

The forest represented by the function $\text{\hspace{0.17em}}B\left(t\right)=82{\left(1.029\right)}^{t}.$

Which forest had a greater number of trees initially? By how many?

Assuming the population growth models continue to represent the growth of the forests, which forest will have a greater number of trees after $\text{\hspace{0.17em}}20\text{\hspace{0.17em}}$ years? By how many?

After $\text{\hspace{0.17em}}t=20\text{\hspace{0.17em}}$ years, forest A will have $\text{\hspace{0.17em}}43\text{\hspace{0.17em}}$ more trees than forest B.

Assuming the population growth models continue to represent the growth of the forests, which forest will have a greater number of trees after $\text{\hspace{0.17em}}100\text{\hspace{0.17em}}$ years? By how many?

Discuss the above results from the previous four exercises. Assuming the population growth models continue to represent the growth of the forests, which forest will have the greater number of trees in the long run? Why? What are some factors that might influence the long-term validity of the exponential growth model?

Answers will vary. Sample response: For a number of years, the population of forest A will increasingly exceed forest B, but because forest B actually grows at a faster rate, the population will eventually become larger than forest A and will remain that way as long as the population growth models hold. Some factors that might influence the long-term validity of the exponential growth model are drought, an epidemic that culls the population, and other environmental and biological factors.

For the following exercises, determine whether the equation represents exponential growth, exponential decay, or neither. Explain.

$y=300{\left(1-t\right)}^{5}$

$y=220{\left(1.06\right)}^{x}$

exponential growth; The growth factor, $\text{\hspace{0.17em}}1.06,$ is greater than $\text{\hspace{0.17em}}1.$

$y=16.5{\left(1.025\right)}^{\frac{1}{x}}$

$y=11,701{\left(0.97\right)}^{t}$

exponential decay; The decay factor, $\text{\hspace{0.17em}}0.97,$ is between $\text{\hspace{0.17em}}0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}1.$

For the following exercises, find the formula for an exponential function that passes through the two points given.

$\left(0,6\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(3,750\right)$

$\left(0,2000\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(2,20\right)$

$f\left(x\right)=2000{\left(0.1\right)}^{x}$

$\left(-1,\frac{3}{2}\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(3,24\right)$

$\left(-2,6\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(3,1\right)$

$f\left(x\right)={\left(\frac{1}{6}\right)}^{-\frac{3}{5}}{\left(\frac{1}{6}\right)}^{\frac{x}{5}}\approx 2.93{\left(0.699\right)}^{x}$

$\left(3,1\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(5,4\right)$

For the following exercises, determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points.

#### Questions & Answers

(1+cosA+IsinA)(1+cosB+isinB)/(cos@+isin@)(cos$+isin$)
Ajay Reply
hatdog
Mark
how we can draw three triangles of distinctly different shapes. All the angles will be cutt off each triangle and placed side by side with vertices touching
Shahid Reply
bsc F. y algebra and trigonometry pepper 2
Aditi Reply
given that x= 3/5 find sin 3x
Adamu Reply
4
DB
remove any signs and collect terms of -2(8a-3b-c)
Joeval Reply
-16a+6b+2c
Will
is that a real answer
Joeval
(x2-2x+8)-4(x2-3x+5)
Ayush Reply
sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
Y
master
X2-2X+8-4X2+12X-20=0 (X2-4X2)+(-2X+12X)+(-20+8)= 0 -3X2+10X-12=0 3X2-10X+12=0 Use quadratic formula To find the answer answer (5±Root11i)/3
master
Soo sorry (5±Root11* i)/3
master
x2-2x+8-4x2+12x-20 x2-4x2-2x+12x+8-20 -3x2+10x-12 now you can find the answer using quadratic
Mukhtar
2x²-6x+1=0
Ife
explain and give four example of hyperbolic function
Lukman Reply
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
Racelle Reply
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Anurag Reply
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
y2=4ax= y=4ax/2. y=2ax
akash
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
Jhovie Reply
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
thanks
Jhovie
What do you call a relation where each element in the domain is related to only one value in the range by some rules?
Charmaine Reply
A banana.
Yaona
given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
Snalo Reply
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Mark Reply
nothing up todat yet
Miranda
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jai
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jai
Miranda Drice
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jai
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Miranda
I am living in india
jai
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Miranda
what is the formula for calculating algebraic
Propessor Reply
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
sita Reply
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
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Propessor
welcome
jai

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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