2-1-25 (1+) Quizizz Study Guide: Exponential Functions

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​Function Transformation

Exponent Rules

Math Response

$$x^5\cdot x^3$$

Type answer here

Correct Answer: x^8

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Math Response

$$\frac{x^5}{x^3}$$

Type answer here

Correct Answer: x^2

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Math Response

$$\frac{x^{-3}\cdot y^2}{x^2}$$

Type answer here

Correct Answer: \frac{y^2}{x^5}

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Math Response

$$\frac{5\left(7x^2\right)^0}{x^{-3}}$$

Type answer here

Correct Answer: 5x^3

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Math Response

$$\left(\frac{2x}{y^2}\right)^{-3}$$

Type answer here

Correct Answer: \frac{y^6}{8x^3}

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Math Response

$$\frac{\left(3a^5r^2\right)^{-1}}{2a^{-5}r^{-2}}$$

Type answer here

Correct Answer: \frac{1}{6}

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​Never leave a perfect square inside the radical.

Math Response

Simplify

$$\sqrt[]{90}$$

Type answer here

Correct Answer: 3\sqrt[2]{10}

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Math Response

$$\sqrt[]{5}\cdot\sqrt[]{10}$$

Type answer here

Correct Answer: 5\sqrt[2]{2}

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Geometric Sequences

Multiple Choice

A certain bacteria population doubles every hour. Suppose initially there is one bacterium. Which of the following is the explicit formula.

Hint: Explicit formula for a geometric sequence: f(n)=f(0)⋅rⁿ

• $$f\left(n\right)=2n+1$$

• $$f\left(n\right)=2^n$$

• $$f\left(n\right)=2\cdot2^n$$

• $$f\left(n\right)=4+2n$$

Multiple Choice

128 players start a tennis tournament. Half of the players advance every round.

Hint: Explicit formula for a geometric sequence: f(n)=f(0)⋅rⁿ

If the sequence is decreasing r is a fraction (a number between 0 and 1)

• $$f\left(n\right)=64\cdot\left(\frac{1}{2}\right)^n$$

• $$f\left(n\right)=128\left(-2\right)^n$$

• $$f\left(n\right)=128\left(\frac{1}{2}\right)^n$$

• $$f\left(n\right)=128-2n$$

Match

Match the description with the graph.

• Exponential Growth

Exponential Growth

Linear Decrease

Linear Increase

Exponential Decay

Quadratic

• Exponential Growth

• Linear Decrease

• Linear Increase

• Exponential Decay

• Quadratic

Multiple Choice

The explicit formula for a geometric sequence is: $$f\left(n\right)=10\left(5\right)^{n-1}$$

Which of the following represent the recursive formula?

Hint:

*Common ratio is the number that has the exponent on it.

*Since the common ratio "5" is raised to a power of n-1, then 10 must be the first term.

*Common ratio tells you that the previous term "f(n-1)" must be multiplied by 5.

• f(1)=2 , f(n)=5f(n-1)

• f(1)=10 , f(n)=5f(n-1)

• f(1)=5 , f(n)=10f(n-1)

• f(1)=10, f(n)=5+n

Multiple Choice

The recursive formula for a geometric sequence is: $$f\left(1\right)=20\ ,\ f\left(n\right)=\frac{1}{4}f\left(n-1\right)$$

Which of the following represents the explicit formula?

Hint:

*First term is 20

*Common ratio is 1/4

*Since we start with the first term, we will raise the common ratio to one less power (n-1).

• $$f\left(n\right)=20\left(\frac{1}{4}\right)^{n-1}$$

• $$f\left(n\right)=20\left(\frac{1}{4}\right)^n$$

• $$f\left(n\right)=\frac{1}{4}\cdot20^{n-1}$$

• $$f\left(n\right)=\frac{1}{4}\cdot20^n$$

Multiple Choice

A certain marine life population initially has 50 members and approximately doubles every 5 years. Which graph represents this species.

Hint:

*Doubles means multiply by 2. Multiplication shows geometric. So exponential curve.

*Numbers are going to grow so the curve must keep increasing from left to right.

*Initial point (y-intercept) must be at 50 .

Drag and Drop

f(x)=​

Placeholder

⋅(

Placeholder

)<sup>x

Hint:

Find common ratio (r): Second term/First term

f(x) = f(0) times r to the power of x</sup>

Drag these tiles and drop them in the correct blank above

0.02

4

0.08

0.32

2

3

Common ratio is greater than 1.

​Exponential Decay

​Exponential Growth

Common ratio is between 0 and 1

Percent Increase

Percent Decrease

Drag and Drop

A new car is purchased for 19900 dollars. The value of the car depreciates at 7% per year. Drag and drop the correct rate that represents the value of the car.

f(x) = 19900 ⋅ (​ ​

Placeholder

)<sup>x

Hint: 100% - 7% = 93%. Write 93% as a decimal, that will be the rate.</sup>

Drag these tiles and drop them in the correct blank above

0.93

1.07

0.7

0.07

1.93

1.7

Fill in the Blanks

A new car is purchased for 19900 dollars. The value of the car depreciates at 7% per year.

^{What is the value of the car after 5 years. Round your answer to the nearest cent.}

^Hint:

^{Nearest cent means nearest hundredth. So, you need to write to numbers after the decimal point.}

Multiple Choice

Which of the following shows an exponential decay (decline)?

• A car depreciates at a rate of 2.2% per year.

• An elevator descends at a rate of 21 feet per second.

• A town's population grows at a rate of 8.9% every year.

• A certain population of 25 aggressive zombies quintuples every month.

Multiple Choice

An element with mass 140 grams decays by 21.6% per minute. Which of the following expressions would correctly show the mass of the element after 20 minutes?

Hint:

At the beginning there is 140 grams.

Decay: 100% - 21.6% = 78.4% will remain. So your rate is 78.4% as a decimal.

20 minutes later so raise to the 20th power.

• $$140\left(0.784\right)^{20}$$

• $$140\left(0.784\right)^{21}$$

• $$140\left(1.216\right)^{20}$$

• $$140\left(0.784\right)^2$$

Graphing

Graph:

$$f\left(x\right)=18\cdot\left(\frac{1}{3}\right)^x$$

Hint:

Number by itself is the y-intercept. Therefore (0,18)

Common ratio has the exponent so: 1/3

So now 18 must be divided by 3 so (1, 6)

Pay attention to the intervals.

Make sure the dashed line (horizontal asymptote remain at x-axis)

Graphing

Graph:

$$f\left(x\right)=2^x$$

Hint:

Think of the equation as f(x)=1⋅2^x

^{so y-intercept is at 1.}

Pay attention to the intervals.

Make sure the dashed line (horizontal asymptote remain at x-axis)

Graphing

Graph:

$$f\left(x\right)=3\cdot2^x$$

Graphing

Graph:

$$f\left(x\right)=15\left(\frac{1}{5}\right)^x$$

Math Response

Write the equation of the function. Write your answer in the form of y=a⋅b^x where a is the zero term and b is the common ratio.

Hint:

Make a table or write the ordered pairs: (0,2), (1,6) , (2,18)

so: 2, 6, 18 then the common ratio is 3.

Zero term is 2.

Type answer here

Correct Answer: y=2\left(3\right)^x

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Math Response

Write the equation of the function. Write your answer in the form of y=a⋅b^x where a is the zero term and b is the common ratio.

Hint:

This is exponential decay. So, the common ratio is between 0 and 1. Write it as a fraction.

Type answer here

Correct Answer: y=12\left(\frac{1}{3}\right)^x

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Solving exponential equations

​Goal is to re-write with the same base.

6^x-1 = 36
Hint: Left side has a base 6. That is the clue which tells you to re-write the right side with base six.
6^x-1 = 6²
Now that the bases equal to each other, this must mean exponents must equal to each other as well.
x-1 = 2
Solve for x. Solution is x=2

Math Response

$$9^{x-4}=3^{x+3}$$

Hint:

9 can be re-written with base 3: 3²

Now that you replaced 9 with 3², the equation will now look like:

$$\left(3^2\right)^{x-4}=3^{x+3}$$

Now you have both sides with the same base.

Set the exponents equal to each other:

$$2\left(x-4\right)=x+3$$

Don't forget to distribute the 2. Solve for x.

Make sure to write your solution in the form of x=....

Type answer here

Correct Answer: x=11

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Math Response

$$4^{2x+2}=256^5$$

Hint: Can you write 256 with base 4?

Type answer here

Correct Answer: x=9

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Math Response

$$9^{2x+5}=27^x$$

Hint:

This is different. You cannot re-write 27 with a base 9.

So, try to write both sides with base 3.

Type answer here

Correct Answer: x=-10

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Math Response

$$2^{3x-3}=\left(\frac{1}{64}\right)$$

Hint:

You can re-write 64 as 2⁶

^{Now, how will you bring that to the top? Think about negative exponents.}

Type answer here

Correct Answer: x=-1

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