# 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.

exercise 1.2 solution b....isnt it lacking
I dnt get dis work well
what is one-to-one function
what is the procedure in solving quadratic equetion at least 6?
Almighty formula or by factorization...or by graphical analysis
Damian
I need to learn this trigonometry from A level.. can anyone help here?
yes am hia
Miiro
tanh2x =2tanhx/1+tanh^2x
cos(a+b)+cos(a-b)/sin(a+b)-sin(a-b)=cotb ... pls some one should help me with this..thanks in anticipation
f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
proof
AUSTINE
sebd me some questions about anything ill solve for yall
cos(a+b)+cos(a-b)/sin(a+b)-sin(a-b)= cotb
favour
how to solve x²=2x+8 factorization?
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
which of these functions is not uniformly cintinuous on (0, 1)? sinx
helo
Akash
hlo
Akash
Hello
Hudheifa
which of these functions is not uniformly continuous on 0,1
solve this equation by completing the square 3x-4x-7=0
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame