Exponential and Logarithmic Graphs

Exponential and Logarithmic Graphs

Growth:
Always Increasing as
x's go up

Heading towards HA as x's go down

Decay:
Always decreasing, heading to HA
as x's go up

as x's go down
going to
infinity

Multiple Choice

Classify the following graph.

• Exponential Growth

• Exponential Decay

• Logarithmic

Multiple Choice

Classify the following graph.

• Exponential Growth
• Exponential Decay
• Logarithmic

​b is the multiplier , when b is bigger than 1 we have exponential growth.

If b is between 0 and 1, then we have exponential decay

​or y intercept

​or y intercept

​y is the ending amount

​y is the ending amount

Multiple Choice

What is y intercept, the initial amount,

for the function:

f(x) = 300(1.16)^x?

• 300
• 1.16
• .16
• x

Multiple Choice

What type of function is

y = 7(5/4)^x?

• Exponential Growth
• Exponential Decay
• Linear
• None of the above

Multiple Choice

Classify the model as Exponential GROWTH or DECAY.
A=1200(.85)⁶

• Growth
• Decay

Multiple Choice

What type of function is

f(x)=2(1/7)^x ?

• Exponential Growth
• Linear
• Exponential Decay
• None of the Abovee

Multiple Choice

In an exponential function,

what does the 'a' represent? 

• SLOPE
• ENDING AMOUNT

• Y-INTERCEPT
• MULTIPLIER

Multiple Choice

$$y=4^{x+5}$$  

What is the transformation from

$$y=4^x$$ ?

• Left 5

• Right 5

• Up 5

• Down 5

Multiple Choice

$$y=4^{x-3}$$  

What is the transformation from

$$y=4^x$$ ?

• Left 3

• Right 3

• Up 3

• Down 3

Multiple Choice

$$y=4^{x+3}-3$$  

What is the transformation from

$$y=4^x$$ ?

• Left 3

  down 3

• Right 3

  down 3

• Right 3

  Up 3

• Left 3

  Up 3

Multiple Choice

$$y=-4^x$$  

What is the transformation from

$$y=4^x$$ ?

• reflected over

  x axis

• reflected over

  y axis

• Right 4

• Down 4

Multiple Choice

What is the Horizontal Asymptote

of

y = 2^x

• y = 0

• y = −3

• x = 0

• x = −3

Multiple Choice

What is the asymptote of y = 2^x - 3

• y = 0

• y = -3

• x = 0

• x = -3

Multiple Choice

What is the Horizontal Asymptote

of y = 2^x +3

• y = 0

• y = 3

• x = 0

• x = 3

Multiple Choice

Where would the asymptote be for this graph?

$$f\left(x\right)=2^{x+3}-4$$  

• y = 4

• y =  −4

• x = 3

• x = −3

The y intercept is where x = 0.

Given an exponential function, set the

x = 0 to find the

y intercept

Multiple Choice

What is the y intercept of

y = 2^x

• (2 , 0)

• (0 , 2)

• (0 , 1)

• (1 , 0)

Multiple Choice

What is the y intercept of

y = 2^x − 3

• (−2, 0)

• (0, −3)

• (0, −2)

• (−3, 0)

Multiple Choice

Find the y intercept for this graph

$$f\left(x\right)=2^{x+3}-4$$  

• (0, 4)

• (0,−4)

• (0, 2)

• (0, 1)

if the x's are going up consistantly,

and the y's go up and there is a constant multiplier

y= 2(3)^x

Multiple Choice

Given the following table of values, construct the function that represents the sequence.

• y = 3(4)^x

• y = 3x + 4

• y = 4(3)^x

• y = 4x + 3

Multiple Choice

Write a function represented by this table?

• f(x) = 36(3)^x

• f(x) = 108(3)^x

• f(x) = 36(2)^x

• f(x) = 108(2)^x

Multiple Choice

Which function represents the table of values? 

• f(x) = 4x
• f(x) = x³
• f(x) = 4^x
• f(x) = 4^x-1

Multiple Choice

Write the function that best models the table?

• y = (1/4)x

• y = 4x

• y = 4^x

• y = 64(1/4)^x

​To solve any equation in Desmos:
Enter each side of the equal sign as y = on a separate line.

Find the point of intersection.
You may have to zoom out several times.

The x of the intersection point is the solutions

Multiple Choice

Solve: 2^x = 4^x+1

• -2
• 2
• -3
• 3

Multiple Choice

7ⁿ⁺¹⁰- 8 = 6

round to 3 places

• -7.374
• -8.643
• -7.360
• -8.853

Multiple Choice

Rewrite 3⁴ = 81 in

logarithmic form.

• log₃4 = 81

• log₈₁3 = 4

• log₃81 = 4

• log₄81 = 3

Multiple Choice

Rewrite in Logarithmic notation

$$3^x=27$$  

• $$\log_327=x$$  

• $$\ln27=x$$  

• $$\log_{27}3=x$$  

• $$\log_x\left(3\right)=27$$

Multiple Choice

Rewrite log₂8 = 3

in exponential form.

• 2⁸ = 3

• 2³ = 8

• 3² = 8

• 8³ = 2

Multiple Choice

Rewrite log₃81 = 4

in exponential form.

• 4³ = 81

• 3⁴ = 81

• 81⁴ = 3

• 3(8)^x = 81

​Exponential graphs always have a horizontal asymptote.


Logarithmic graphs always have a vertical asympote

In this case the vertical asymptote is x = 0

If there is no number added or subtracted from the x, the vertical aysmptote will be x = 0

​Exponential graphs always have a horizontal asymptote.


Logarithmic graphs always have a vertical asympote


Notice how the orange graph
f(x) = log₂(x) has a vertical aysmptote of x = 0

While the blue graph
f(x) = log₂ (x + 1)
has shifter left 1, so the VA is x = -1

Multiple Choice

Which of the following has a vertical asymptote at $$x=4$$  

Hint: exponential functions have horizonal aysmptotes

• $$y=2^x-4$$  

• $$y=\log_2\left(x-4\right)$$  

• $$y=\log_2\left(x+4\right)$$  

Multiple Choice

Logarithmic functions are the inverse of...

• Linear Functions

• Exponential Functions

• Quadratic Functions

• Polynomial Functions

​If it is a multiple choice question, you can also use Desmos to determine the inverse.

Multiple Choice

Find the inverse of the function:

f(x) = 10^3x

• f^-1(x) = log(3x)

• f^-1(x) = 3log(x)

• f^-1(x) = ⅓ log(x)

Multiple Choice

Which is the inverse for the function?

f(x) = 2^2x-3

• f^-1(x) = log₂(x) − 6

• f^-1(x) = log₂(x) + 6

• f^-1(x)=½log₂(x) + 3/2

​y is the ending amount
a is the starting amount
r is the interest rate % as a decimal (divided by 100)
n is the number of times interest is applied in a year
t is the number of years

​Compound Interest

This is used when interest is applied more than once a year

Multiple Choice

What does the n stand for in this formula?

• Initial amount

• Final amount

• Rate

• Time

• The number of times compounded in 1 year

Multiple Choice

What does the t stand for in this formula?

• Initial amount

• Final amount

• Rate

• Time, numbet of

  years

• The number of times compounded per year

Multiple Choice

What does the a stand for in this formula?

• Initial amount

• Final amount

• Rate

• Time, numbet of

  years

• The number of times compounded per year

Multiple Choice

What is the formula for annual exponential growth?

• $$f\left(x\right)=a\left(1-r\right)^x$$

• $$f\left(x\right)=a\left(1+r\right)^x$$

Multiple Choice

What is the formula for annual exponential decay?

• $$f\left(x\right)=a\left(1-r\right)^x$$

• $$f\left(x\right)=a\left(1+r\right)^x$$

Multiple Choice

Exponential growth is  $$y=a\left(1+r\right)^x$$  , and exponential decay is  $$y=a\left(1-r\right)^x$$  .

What does "a" represent?

• Initial amount

• Rate of growth or decay

• Time

• End amount

Multiple Choice

Exponential growth is  $$y=a\left(1+r\right)^x$$  , and exponential decay is  $$y=a\left(1-r\right)^x$$  .

What does "r" represent?

• Initial amount

• Rate of growth or decay as a decimal

• Time

• End amount

Multiple Choice

Convert the percent to a decimal. Remember: move the decimal two places to the left!

3.5%=

• .35

• 3.5

• .035

• .0035

Multiple Choice

Convert the percent to a decimal:

27%=

• 2.7

• .27

• 270

• .027

Multiple Choice

Convert the percent to a decimal:

9%=

• .9

• 900

• .009

• .09

Multiple Choice

Which of the following functions shows an initial amount of

$15 and an increase of 35% each year?

• y = 15(35)^x
• y = 15(1.35)^x
• y = 15(0.35)^x
• y = 35(1.15)^x

Multiple Choice

The value of a car is $15,000 and depreciates at a

rate of 8% per year.

What is the exponential equation?

• y = 15,000(1.08)^x

• y = 15,000(0.92)^x

• y = 15,000(.08)^x

Multiple Choice

Suppose a culture of bacteria begins with 5000 cells and

dies by 30% each year.

Write an equation that represents this situation.

• y = 5000(0.7)^x

• y = 30(5000)^x

• y = 5000(1.3)^x

• y = 5000x^x

Multiple Choice

Write an equation that models:
Samantha's hair was known to grow very rapidly.

It began at a length of 6 in and grew at a rate of 14% a week.

• y = 6(0.14)^x

• y = 6(1+14)^x

• y = 6(1.14)^x

• y = 6(0.86)^x

​to find the rate %

Subtract 1 from the multiplier and multiply times 100

1.25 - 1 = .25 (100) = 25%

​to find the rate %

Subtract 1 from the multiplier and multiply times 100

​.54 - 1 = -.46
-.46 (100) decrease by 46%

​.54 - 1 = -.46
-.46 (100) decrease by 46%


1.35 - 1 = .35
.35(100) = increase by 35%

​.54 - 1 = -.46
-.46 (100) decrease by 46%


1.35 - 1 = .35
.35(100) = increase by 35%


.67 - 1 = -.33
-.33(100) = decrease by 33%

Multiple Choice

Find the increase or decrease %

y = 3(0.85)^x

• 15% increase

• 15% decrease

• 85% increase

• 85% decrease

Multiple Choice

Find the increase or decrease %

y = 3(1.25)^x

• 25% increase

• 25% decrease

• 125% increase

• 125% decrease

Multiple Choice

When do you use this formula? Select all

• used when compounded more than once a year

• used when compounded continously

• when given the multiplier with a word like double or triple

• used when increasing or decreasing by a % rate

Multiple Choice

A particular type of cell triples in number every hour.

Which function can be used to find the number of cells present

at the end of h hours if there are initially 6 of these cells?

• f(h) = 3(6)^h

• f(h) = 6x^h

• f(h) = 6(3)^h

• f(h) = 3(6h)

Multiple Choice

A tennis tournament begins with 128 players. At the end of round one,

there will be 64 players. There are 32 players at the end of round 2 and 16

players remaining after round 3.

Write the equation that best models this situation.

• y = −64x + 128

• y = 0.5x + 128

• y = 128(0.5)^x

• y = 128(2)^x