Worksheet: Writing Exponential Functions

In this worksheet, we will practice writing equivalent equations for exponential functions using tables of values, graphs, and descriptions.

Q1:

Write an exponential equation in the form 𝑦 = 𝑏  for the numbers in the table.

π‘₯ 2 4 5
𝑦 9 1 6 8 1 2 5 6 2 4 3 1 0 2 4
  • A 𝑦 = ( π‘₯ )  
  • B 𝑦 = ο€Ό 9 3 2  
  • C 𝑦 = ( π‘₯ )   οŽ₯
  • D 𝑦 = ο€Ό 9 1 6  
  • E 𝑦 = ο€Ό 3 4  

Q2:

Write an exponential equation in the form 𝑦 = π‘Ž ( 𝑏 )  for the numbers in the table.

π‘₯ 0 1 2 3
𝑦 18 6 2 2 3
  • A 𝑦 = 2 ( 3 ) 
  • B 𝑦 = 3 ( 2 ) 
  • C 𝑦 = 2 π‘₯ 
  • D 𝑦 = 1 3 ( 1 8 ) 
  • E 𝑦 = 1 8 ο€Ό 1 3  

Q3:

The curve below is 𝑦 = 𝐴 βˆ’ 𝐡 ( 1 βˆ’ 𝑏 )   for positive constants 𝐴 , 𝐡 , and 𝑏 . Determine these, in that order, using the information in the figure.

  • A 𝐴 = 1 5 , 𝐡 = 1 0 , 𝑏 = √ 2
  • B 𝐴 = 1 5 , 𝐡 = 6 , 𝑏 = 2
  • C 𝐴 = 1 0 , 𝐡 = 5 , 𝑏 = 1 6
  • D 𝐴 = 1 0 , 𝐡 = 5 , 𝑏 = √ 2 
  • E 𝐴 = 1 5 , 𝐡 = 4 , 𝑏 = 4

Q4:

An initial bacteria population 𝑃  doubles every hour, which is described by the formula 𝑃 = 𝑃 β‹… 2   . Write 𝑃 in the form 𝑃 = 𝑃 β‹… 𝑏     stating 𝑏 in scientific notation, to two significant figures, and find the number by which you would need to multiply the first days population to find the second.

  • A 𝑃 = 𝑃 β‹… ο€Ή 1 . 7 β‹… 1 0     , the bacteria population is multiplied by 1 . 7 β‹… 1 0  every hour
  • B 𝑃 = ο€Ή 1 . 7 β‹… 1 0      , the bacteria population is multiplied by 1 . 7 β‹… 1 0  every day
  • C 𝑃 = 𝑃 β‹… ο€Ή 1 . 7 β‹… 1 0       , the bacteria population is multiplied by 1 . 7 β‹… 1 0  every day
  • D 𝑃 = 𝑃 β‹… ο€Ή 1 . 7 β‹… 1 0       , the bacteria population is multiplied by 1 . 7 β‹… 1 0  every day
  • E 𝑃 = 𝑃 β‹… ( 2 )     , the bacteria population is multiplied by 2 every day

Q5:

The producer of a successful radio show predicts that the number of listeners will increase by 0.5% a month. The show currently has 4 5 0 0 0 listeners. Write an equation that can be used to calculate 𝐿 , the number of listeners they expect to have in 𝑑 years’ time.

  • A 𝐿 = 4 5 0 0 0 ( 1 . 0 0 5 )   
  • B 𝐿 = 4 5 0 0 0 ( 0 . 9 9 5 )   
  • C 𝐿 = 4 5 0 0 0 ( 0 . 0 0 5 )   
  • D 𝐿 = 4 5 0 0 0 ( 0 . 9 9 5 )   
  • E 𝐿 = 4 5 0 0 0 ( 1 . 0 0 5 )   

Q6:

Write an equation in the form 𝑦 = π‘Ž ( 𝑏 )  for the numbers in the table.

π‘₯ 1 3 4
𝑦 24 6 βˆ’ 3
  • A 𝑦 = 4 8 ο€Ό 1 2  
  • B 𝑦 = βˆ’ 4 8 ο€Ό βˆ’ 1 2  
  • C 𝑦 = βˆ’ 1 2 ( βˆ’ 2 ) 
  • D 𝑦 = 1 2 π‘₯ 
  • E 𝑦 = 1 2 ( 2 ) 

Q7:

Write an exponential equation in the form 𝑦 = 𝑏  for the numbers in the table.

π‘₯ 0 1 2 3
𝑦 1 5 25 125
  • A 𝑦 = 2 5 
  • B 𝑦 = π‘₯ 
  • C 𝑦 = π‘₯  
  • D 𝑦 = 5 
  • E 𝑦 = 5  

Q8:

Write an exponential equation in the form 𝑦 = π‘Ž ( 𝑏 )  for the numbers in the table.

π‘₯ 0 1 2 3
𝑦 βˆ’ 2 βˆ’ 1 0 βˆ’ 5 0 βˆ’ 2 5 0
  • A 𝑦 = βˆ’ 2 ( 5 ) 
  • B 𝑦 = 5 ( βˆ’ 2 ) 
  • C 𝑦 = 2 ( βˆ’ 5 ) 
  • D 𝑦 = βˆ’ 5 ( 2 ) 
  • E 𝑦 = βˆ’ 2 π‘₯ 

Q9:

Observe the given graph, and then answer the following questions.

Find the 𝑦 -intercept in the shown graph.

As this graph represents an exponential function, every 𝑦 -value is multiplied by 𝑏 when π‘₯ increases by Ξ” π‘₯ . Find 𝑏 for Ξ” π‘₯ = 1 .

Find the equation that describes the graph in the form 𝑦 = π‘Ž 𝑏    .

  • A 𝑦 = 1 0 β‹… 2 
  • B 𝑦 = 1 0 β‹… ο€Ό 1 2  
  • C 𝑦 = 2 
  • D 𝑦 = 1 0 β‹… 4 
  • E 𝑦 = 1 0 β‹… ο€Ό 1 4  

Q10:

The given graph shows that 𝑦 = 𝑓 ( π‘₯ ) .

Write an explicit formula for 𝑓 ( π‘₯ ) in the form 𝑓 ( π‘₯ ) = π‘Ž 𝑏   .

  • A 𝑓 ( π‘₯ ) = 2 β‹… 3  
  • B 𝑓 ( π‘₯ ) = 2 β‹… 2  
  • C 𝑓 ( π‘₯ ) = 3 β‹… 2  
  • D 𝑓 ( π‘₯ ) = 2 β‹… 2 
  • E 𝑓 ( π‘₯ ) = 2 β‹… 2  

Graphically estimate the number that 𝑦 is multiplied by when π‘₯ increases by 1.

What calculation would allow you to find an accurate value for 𝑏 if you wanted to write 𝑓 ( π‘₯ ) in the form 𝑓 ( π‘₯ ) = π‘Ž 𝑏  ?

  • A 3 = √ 3  
  • B 2 = √ 2  
  • C 2 = √ 2   

Q11:

Observe the given graph, and then answer the following questions.

Find the 𝑦 -intercept in the shown graph.

As this graph represents an exponential function, every 𝑦 -value is multiplied by 𝑏 when π‘₯ increases by Ξ” π‘₯ . Find 𝑏 for Ξ” π‘₯ = 1 .

  • A 1 4
  • B4
  • C 1 2
  • D2
  • E 1 3

Find the equation that describes the graph in the form 𝑦 = π‘Ž 𝑏    .

  • A 𝑦 = 4 0 0 β‹… ο€Ό 1 2  
  • B 𝑦 = 4 0 0 β‹… ο€Ό 1 4  
  • C 𝑦 = 4 0 0 β‹… π‘₯ οŠͺ
  • D 𝑦 = 4 0 0 β‹… ο€Ό 1 3  
  • E 𝑦 = 4 β‹… ο€Ό 1 4 0 0  

Q12:

Observe the given graph, and then answer the following questions.

Find the 𝑦 -intercept in the shown graph.

As this graph represents an exponential function, every 𝑦 –value is multiplied by 𝑏 when π‘₯ increases by Ξ” π‘₯ . Find 𝑏 for Ξ” π‘₯ = 3 .

  • A 1 5
  • B4
  • C 1 3
  • D 1 4
  • E5

Find the equation that describes the graph in the form 𝑦 = π‘Ž 𝑏 π‘₯ Ξ” π‘₯ .

  • A 𝑦 = 2 5 0 0 β‹… ο€Ό 1 2  π‘₯ 4
  • B 𝑦 = 2 5 0 0 β‹… ο€Ό 1 5  π‘₯ 3
  • C 𝑦 = 2 5 0 0 β‹… ο€Ό 1 5  π‘₯
  • D 𝑦 = 2 5 0 0 β‹… ο€Ό 1 4  π‘₯ 2
  • E 𝑦 = 2 5 0 0 β‹… ο€Ό 1 5  3 π‘₯

Q13:

The given graph shows that 𝑦 = 𝑓 ( π‘₯ ) .

Write an explicit formula for 𝑓 ( π‘₯ ) in the form 𝑓 ( π‘₯ ) = π‘Ž 𝑏   .

  • A 𝑓 ( π‘₯ ) = 6 β‹… ο€Ό 1 3   
  • B 𝑓 ( π‘₯ ) = 6 β‹… ο€Ό 1 3  
  • C 𝑓 ( π‘₯ ) = 6 β‹… ο€Ό 1 5   
  • D 𝑓 ( π‘₯ ) = 6 β‹… 3  
  • E 𝑓 ( π‘₯ ) = 6 β‹… ο€Ό 1 3   

Graphically estimate the number that 𝑦 is multiplied by when π‘₯ increases by 1.

  • A0.8
  • B0.03
  • C0.3
  • D4.9
  • E3

What calculation would allow you to find an accurate value for 𝑏 if you wanted to write 𝑓 ( π‘₯ ) in the form 𝑓 ( π‘₯ ) = π‘Ž 𝑏  ?

  • A ( 3 ) = 2 4 3 
  • B ( 3 ) = √ 3   
  • C ( 5 ) = √ 5   
  • D ο€Ό 1 3  = ο„ž 1 3   
  • E ο€Ό 1 5  = ο„ž 1 5   

Q14:

The number of people who view a meme triples every hour. If 5 friends viewed a meme initially, how many people would have viewed it after one hour?

How many people would have viewed the meme after 𝑑 hours?

  • A 5 ( 3 ) 
  • B 5 ( 2 ) 
  • C 1 5 
  • D 3 ( 5 ) 
  • E 5 ( 𝑑 ) 

Q15:

What must a number be multiplied by to increase it by 13%?

Write an equation to represent the statement β€œTo calculate the value of 𝑦 , increase π‘₯ by 13%.”

  • A 𝑦 = 0 . 1 3 π‘₯
  • B 𝑦 = 1 . 0 1 3 π‘₯
  • C 𝑦 = 0 . 8 7 π‘₯
  • D 𝑦 = 1 . 1 3 π‘₯
  • E 𝑦 = 1 3 π‘₯

A company aims to increase its profits by 13% every year for the next 3 years. If its profit this year is 𝑃  , write an equation to calculate 𝑃  , the profit in 3 years’ time.

  • A 𝑃 = 𝑃 ( 1 . 0 1 3 )   
  • B 𝑃 = 𝑃 ( 0 . 8 7 )   
  • C 𝑃 = 𝑃 ( 1 3 )   
  • D 𝑃 = 𝑃 ( 0 . 1 3 )   
  • E 𝑃 = 𝑃 ( 1 . 1 3 )   

Q16:

What must a number be multiplied by to decrease it by 5%?

Write an equation to represent the statement β€œTo calculate the value of 𝑦 , decrease π‘₯ by 5%.”

  • A 𝑦 = 0 . 0 5 π‘₯
  • B 𝑦 = 1 . 0 5 π‘₯
  • C 𝑦 = 1 . 5 π‘₯
  • D 𝑦 = 0 . 9 5 π‘₯
  • E 𝑦 = 0 . 5 π‘₯

A manufacturer aims to decrease the amount of waste they produce by 5% every year. If they currently produce 45 tons of waste, write an equation to calculate π‘Š , the amount they aim to produce in 𝑑 years .

  • A π‘Š = 4 5 ( 0 . 0 5 ) 
  • B π‘Š = 4 5 ( 0 . 5 ) 
  • C π‘Š = 4 5 ( 0 . 9 5 ) 
  • D π‘Š = 4 5 ( 1 . 0 5 ) 
  • E π‘Š = 4 5 ( 1 . 5 ) 

Q17:

Olivia bought an antique vase for $600. The value of the vase increases by 4% each year. Write an equation that can be used to find the value of the vase in dollars, 𝐴 , 𝑑 years after it was purchased.

  • A 𝐴 = 6 0 0 ( 0 . 0 4 ) 
  • B 𝐴 = 6 0 0 ( 1 . 0 4 ) 
  • C 𝐴 = 6 0 0 ( 1 . 4 ) 
  • D 𝐴 = 6 0 0 ( 4 ) 
  • E 𝐴 = 6 0 0 ( 0 . 9 6 ) 

Q18:

Ethan decided to invest his savings in a high-interest account that pays 7% interest compounded annually. He invests $450 in the account for 𝑑 years and makes no other deposits or withdrawals.

Write an equation that can be used to calculate 𝑉 , the value of his investment in dollars after 𝑑 years.

  • A 𝑉 = 4 5 0 ( 𝑑 )  οŽ–  
  • B 𝑉 = 4 5 0 ( 0 . 9 3 ) 
  • C 𝑉 = 4 5 0 ( 0 . 0 7 ) 
  • D 𝑉 = 4 5 0 ( 𝑑 )  οŽ–  
  • E 𝑉 = 4 5 0 ( 1 . 0 7 ) 

What will be the value of his investment after 7 years? Give your answer to the nearest dollar.

Emma invested her savings in the same account 5 years ago. She made no further deposits or withdrawals. If the balance in her account is now $ 1 , 0 2 0 , find, to the nearest dollar, her initial deposit.

Q19:

David wants to invest his savings. He has found a fund which pays 7 % a year, compounded annually. He would like to have $ 1 2 , 0 0 0 in that fund after 3 years. Write an equation that can be used to find 𝑃 , the amount in dollars that David must invest for him to have $ 1 2 , 0 0 0 in the fund after 3 years.

  • A 𝑃 = 1 2 0 ( 1 . 0 7 ) 
  • B 𝑃 = 1 2 , 0 0 0 ( 1 . 0 7 ) 
  • C 𝑃 = 1 2 0 ( 0 . 0 7 ) 
  • D 𝑃 ( 0 . 0 7 ) = 1 2 , 0 0 0 
  • E 𝑃 ( 1 . 0 7 ) = 1 2 , 0 0 0 

Q20:

Write an exponential equation in the form 𝑦 = π‘Ž ( 𝑏 )  for the numbers in the table.

π‘₯ 0 1 2 3
𝑦 5 15 45 135
  • A 𝑦 = 5 ( 3 ) 
  • B 𝑦 = 5 π‘₯ 
  • C 𝑦 = 3 ( 5 ) 
  • D 𝑦 = 1 5 
  • E 𝑦 = 3 π‘₯ 

Q21:

Write an equation in the form 𝑦 = π‘Ž ( 𝑏 )  for the numbers in the table.

π‘₯ 0 1 2 3
𝑦 βˆ’ 4 6 βˆ’ 9 13.5
  • A 𝑦 = 4 ( βˆ’ 1 . 5 ) 
  • B 𝑦 = βˆ’ 1 . 5 ( 4 ) 
  • C 𝑦 = βˆ’ 1 . 5 ( βˆ’ 4 ) 
  • D 𝑦 = βˆ’ 4 ( βˆ’ 1 . 5 ) 
  • E 𝑦 = βˆ’ 4 π‘₯   οŽ– 

Q22:

Write an equation to represent the statement β€œThe value of 𝑦 is equal to five to the power of π‘₯ .”

  • A 𝑦 = π‘₯ 
  • B 𝑦 = 5 π‘₯
  • C 𝑦 = π‘₯  
  • D 𝑦 = 5  
  • E 𝑦 = 5 

Q23:

Write an exponential equation in the form 𝑦 = π‘Ž ( 𝑏 )  for the numbers in the table.

π‘₯ 1 2 3
𝑦 4 3 1 6 1 5 6 4 7 5
  • A 𝑦 = 4 5 ο€Ό 5 3  
  • B 𝑦 = 5 3 ο€Ό 4 5  
  • C 𝑦 = 1 6 1 5 ο€Ό 5 4  
  • D 𝑦 = 5 3 ο€Ό 5 4  
  • E 𝑦 = 5 4 ο€Ό 1 6 1 5  

Q24:

Write an exponential equation in the form 𝑦 = 𝑏  for the numbers in the table.

π‘₯ 0 1 2 3
𝑦 1 2 5 4 2 5 8 1 2 5
  • A 𝑦 = ο€Ό 2 5  
  • B 𝑦 = ο€Ό 4 2 5  
  • C 𝑦 = ( π‘₯ )   
  • D 𝑦 = ο€Ό 2 2 5  
  • E 𝑦 = ( π‘₯ )  

Q25:

Write an equation to represent the statement β€œWhen 1.5 is raised to the power of π‘₯ and the result is multiplied by 4, the answer is 𝑦 .”

  • A 𝑦 = 1 . 5 ( 4 ) 
  • B 𝑦 = 6 
  • C 𝑦 = 4 ( 1 . 5 ) 
  • D 𝑦 = 4 ( π‘₯ )  οŽ– 
  • E 𝑦 = 1 . 5 ( π‘₯ ) οŠͺ

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