Exponential Functions

Exponential Functions

By Shalynne Orth

Exponential Functions have a variable in the exponent

Multiple Choice

Which of the following is an exponential function?

• y = x²

• f(x) = 3x³

• y = x³

• f(x) = 3^x

Exponential Functions

f(x) = a (b)^x

^{a = initial value or starting amount}

^{b= (common ratio) growth or decay factor}

Multiple Choice

What is the initial value in y=4(2)^x

• 4

• 2

• x

Multiple Choice

What is the y-intercept in y=4(2)^x

• (4,0)

• (0,2)

• x

• (0,4)

• (4,2)

Multiple Choice

What is the starting amount in the problem below?

f(t) = 300(1.05)^t

• 300

• 1.05

• t

• y

Multiple Choice

What is the growth rate in the problem below?

f(t) = 300(1.05)^t

• 300

• 1.05

• t

• y

Multiple Choice

What is the common ratio for the function f(x) = 25 (4)^x

• 25

• 4

Multiple Choice

The y-intercept for any function in the form

y=b^x

• (0,0)

• (0,1)

• (1,0)

Multiple Choice

What is the y-intercept:

$$y=3^x$$  

• 0

• 1

• 3

• 6

Multiple Choice

James' 70 in. giant peach doubles in size every week. Write an expression that would represent how big the peach is after 5 weeks.

• 70(2)³⁵
• 70(2)⁵
• 2(70)⁵
• 5(70)²

Exponential Growth vs. Decay

If b > 1, then Growth

If 0 < b < 1, then Decay

Both Graphs have a

Horizontal Asymptote

(graph never touches

the X-axis)

b^x cannot equal 0

Multiple Choice

Classify the following graph.

• Exponential Growth

• Exponential Decay

• Logarithmic

Multiple Choice

Classify the following graph.

• Exponential Growth

• Exponential Decay

• Logarithmic

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

Graphing Exponential Equations

Multiple Choice

What equation matches the graph?

• f(x) = 3^x

• f(x) = (⅓)^x

• f(x) = log₃(x)

Multiple Choice

Which equation is representative of the graph?

• y=2(3)^x

• y=3(2)^x

• y=2(6)^x

Multiple Choice

Which equation is representative of the graph?

• y=4(1/2)^x

• y=1/2(4)^x

• y=4(2)^x

• y=8(1/2)^x

Asymptote

In mathematics, an asymptote is a horizontal, vertical, or slanted line that a graph approaches but never touches.

The asymptote of an exponential function is a horizontal line.

The equation of a horizontal asymptote can be written as

y = constant.

Aysmptote of the Exponential parent function

The asymptote of the exponential parent function is the horizontal line y = 0.

Multiple Choice

Vertical shifts work the same way in exponential functions as they do in quadratics. The number added or subtracted at the end is the vertical shift - and in this case, the asymptote!

• y = 0

• y = 5

• y = 1

• All real numbers

Multiple Choice

Vertical shifts work the same way in exponential functions as they do in quadratics. The number added or subtracted at the end is the vertical shift - and in this case, the asymptote!

• y = 2

• y = 0

• y = 6

• all real numbers

Multiple Choice

What is the range of the function?

• All real numbers

• y > 3

• y < 4

• Impossible to determine from the graph.

Two types of exponential situations

- Decay

-Growth

Growth Formula

Decay Formula

A 97 mg sample of a radioactive substance is decreasing at a rate of 5% per hour. How much will be left after 6 hours?

• a = Plug in 97 milligrams for the initial amount

• r = 0.05 for the rate of decrease

• t = 6 hours for the time elapsed.

A colony of 300 bacteria doubles in size every 43 minutes. What will the population be 86 minutes from now?

• First, find out how many times the population will double. Divide the number of minutes by how long it takes for the population to double.  86 ÷ 43 = 2

• Now figure out what the population will be after it doubles 2 times. Multiply the population by 2 a total of 2 times.

• 300 × 2 × 2 = 1,200

• After 86 minutes, the population will be 1,200 bacteria

Multiple Choice

Write an equation that models the following situation:
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

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=8(15,000)^x

• y=15,000(1 + .08)^x

  or y=15,000(1.08)^x

• y=15,000(0.92)^x

  or y=15,000(1 - .08)^x

• y=15,000(0.08)^x

Fill in the Blanks

A mountain in Scott County is eroding and losing elevation at a rate of 5% every millennium. If the current elevation is 690 meters, how tall will the mountain be in 7 millennia?

If necessary, round your answer to the nearest whole number. (only type number)

Fill in the Blanks

A town's population is currently 2,500. If the population doubles every 57 years, what will the population be 171 years from now? (just type number)

Fill in the Blanks

The half-life of a radioactive kind of curium is 18 years. How much will be left after 72 years, if you start with 608 grams of it? (only type number)

Fill in the Blanks

This year, Hillsdale recycled 52,461 tons of material. If this amount increases by 7% each year, how much material will be recycled 4 years in the future? If necessary, round your answer to the nearest tenth.

Multiple Choice

Linear or Exponential:

$$y=2x+7$$  

• Linear

• Exponential

Multiple Choice

Linear or Exponential:

$$y=2^x+6$$  

• Linear

• Exponential

Multiple Choice

Is the table linear or exponential?

• Linear

• Exponential

Multiple Choice

Is the table linear or exponential?

• Linear

• Exponential

Multiple Choice

The graph of this function is a straight line.

• Linear

• Exponential

• Neither

Multiple Choice

The graph of this function is a curve going up really fast or down really fast. 

• Linear 
• Exponential
• Neither

Multiple Choice

Is this graph linear or exponential?

• Linear

• Exponential

• Neither

Multiple Choice

Is this graph linear or exponential?

• Linear

• Exponential

• Neither

Multiple Choice

What type of function describes the story below?
You have $5 and triple your money every day. 

• Linear
• Exponential
• Neither

Multiple Choice

What type of function describes the story below?
You have $10 and earn $5 every week. 

• Linear
• Exponential
• Neither