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GCSE Maths Algebra Functions

Exponential Function

Exponential Function

Here we will learn about the exponential function, graphs of exponential functions, finding exponential equations and its links to compound interest and geometric sequences.

There are also exponential function worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is an exponential function?

An exponential function is is a mathematical function in the form y=ab^x, where x and y are variables, and a and b are constants, b > 0.

For example,

The diagram shows the graphs of y=2^x, \, y=0.4^x, and y=0.5(3^x).

Exponential Function image 1

The graph of an exponential function has a horizontal asymptote. These all have a horizontal asymptote at y=0 (the x -axis) because ab^x can never equal zero.

Exponential functions are a very important part of mathematics and can be applied to many real-life situations such as finance, population growth, radioactive decay, and the spread of viruses.

If something is said to have exponential growth or exponential decay, then it can be modelled using an exponential function.

The word “exponent” means index or power. So an equation in the form y=ab^x is called an exponential equation because the variable x is the power in the equation.

A exponential function can also have a negative exponent y=ab^{-x}.

These can be rewritten with a positive exponent using the laws of indices.

For example,

y=4^{-x}

y=(4^{-1})^x

y=\left( \cfrac{1}{4}\right)^x

Exponential Function image 2

In general,

y=ab^{-x} is the same as y=a\left(\cfrac{1}{b}\right)^x .

Note that b will always be positive.

What is an exponential function?

What is an exponential function?

How to draw exponential graphs

In order to draw exponential graphs:

  1. Make a table of values for \textbf{x} and calculate the \textbf{y} -values.
  2. Plot the points on an \textbf{xy} coordinate plane
  3. Join the points with a smooth curve.

Explain how to draw exponential graphs

Explain how to draw exponential graphs

Exponential function worksheet

Get your free exponential function worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Exponential function worksheet

Get your free exponential function worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOON

Drawing exponential graph examples

Example 1: drawing an exponential graph in the form y = ax, a > 1

Draw the graph of y=3^x on the axes provided.

Exponential Function example 1 image 1

  1. Make a table of values for \textbf{x} and calculate the \textbf{y} -values.

We have x values from -2 to 3.

We can find the y -values with or without a calculator.

Exponential Function example 1 image 2

2Plot the points on an \textbf{xy} coordinate plane.

Exponential Function example 1 image 3

3Join the points with a smooth curve.

Exponential Function example 1 image 4

Example 2: drawing an exponential graph in the form y = ax, 0 < a < 1

Draw the graph of y=0.5^x on the axes provided.

Exponential Function example 2 image 1

We have x values from -3 to 2.


We can find the y -values with or without a calculator.


Exponential Function example 2 image 2

Exponential Function example 2 image 3

Exponential Function example 2 image 4

Example 3: drawing an exponential graph in the form y = abx

Draw the graph of y=0.2(4^x) on the axes provided.

Exponential Function example 3 image 1

We have x values from -2 to 2.


We can find the y -values with or without a calculator.


Exponential Function example 3 image 2

Exponential Function example 3 image 3

Exponential Function example 3 image 4

How to find exponential functions

In order to find exponential functions:

  1. Find two points that lie on the graph. 
  2. Form two equations in the form \bf{y=ab^x}.
  3. Solve the equations simultaneously.

Explain how to find exponential functions

Explain how to find exponential functions

Finding exponential functions examples

Example 4: finding exponential functions given the y-intercept

Find the equation of the exponential function in the form y=ab^x.

Exponential Function example 4

We have the points (0, 2) and (1, 5).

(0, 2) gives us 2=ab^0.


(1, 5) gives us 5=ab^1.

We can simplify the two equations before solving.


b^0=1, therefore 2=ab^0 gives us a=2.


b^1=b, therefore 5=ab^1 gives us 5=ab.


We know a=2, therefore

5 = 2b

b=2.5 .


The exponential equation is y=2(2.5^x).

Example 5: finding exponential functions by dividing the equations

Find the equation of the exponential function in the form y=ab^x .

Exponential Function example 5

We have the points (1, 4.5) and (2, 13.5).

(1, 4.5) gives us 4.5=ab.


(2, 13.5) gives us 13.5=ab^{2}.

We can solve the equations simultaneously by dividing one equation by the other.


\cfrac{13.5}{4.5}=\cfrac{ab^2}{ab}


This gives 3=b.


Substituting this into 4.5=ab gives


4.5=3a


a=1.5


The exponential equation is y=1.5(3^x).

Example 6: finding exponential functions by dividing the equations

Find the equation of the exponential function in the form y=ab^x .

Exponential Function example 6

We have the points (-4, 80) and (-1, 10).

(-4, 80) gives us 80=ab^{-4}.


(-1, 10) gives us 10=ab^{-1}.

We can solve the equations simultaneously by dividing one equation by the other.


\cfrac{10}{80}=\cfrac{ab^{-1}}{ab^{-4}}


This gives


\cfrac{1}{8}=b^3


\cfrac{1}{2}=b


Substituting this into 10=ab^{-1} gives


10=2a


a=5


The exponential equation is y=5(0.5^x).

Common misconceptions

  • Applying the exponent incorrectly

A common error is to confuse y=ab^x with y=(ab)^x. This is especially important when substituting values into a calculator or finding values to plot a graph. y=ab^x means y=a\times (b^x).

  • Confusing exponentials with polynomials

Some students can make the mistake of thinking 2^x is the same as x^2 and calculate 2^3 as 9.

Practice exponential function questions

1. Which of the graphs shows y=2^x?

Exponential Function practice question 1 image 1

GCSE Quiz False

Exponential Function practice question 1 image 2

GCSE Quiz True

Exponential Function practice question 1 image 3

GCSE Quiz False

Exponential Function practice question 1 image 4

GCSE Quiz False

y=2^x has the table of values,

 

Exponential Function practice question 1 image 5

2. Which of the graphs is y=\left(\cfrac{1}{2}\right)^x?

 

Exponential Function practice question 2 image 1

A

GCSE Quiz False

B

GCSE Quiz False

C

GCSE Quiz False

D

GCSE Quiz True

y=\bigl(\cfrac{1}{2}\bigr)^x has the table of values,

 

Exponential Function practice question 2 image 2

3. Which of the graphs is y=2\left(3^{-x}\right)?

 

Exponential Function practice question 3 image 1

A

GCSE Quiz True

B

GCSE Quiz False

C

GCSE Quiz False

D

GCSE Quiz False

y=2\left(3^{-x}\right) has the table of values,

 

Exponential Function practice question 3 image 2

4. An exponential function goes through the points (0, 1) and (2, 25). What is the function?

y=5^x
GCSE Quiz True

y=x^5
GCSE Quiz False

y=25^x
GCSE Quiz False

y=12.5x
GCSE Quiz False

Using the points and y=ab^x, \, b>0.

 

(0, 1) gives 1=a.

 

(2, 25) gives 25=ab^2.

 

b^2=25

 

b=5, because b is always positive.

5. An exponential function goes through the points (0, 3) and (2, 48). What is the function?

y=4(3^x)
GCSE Quiz False

y=3(4^x)
GCSE Quiz True

y=3(2^x)
GCSE Quiz False

y=4^x
GCSE Quiz False

Using the points and y=ab^x, \, b>0.

 

(0, 3) gives 3=a.

 

(2, 48) gives 48=ab^2.

 

48=3b^2

 

b^2=16

 

b=4, because b is always positive.

6. An exponential function goes through the points (1, 9) and (4, 30.375). What is the function?

y=2.5(6^x)
GCSE Quiz False

y=1.5(6^x)
GCSE Quiz False

y=6(2.5^x)
GCSE Quiz False

y=6(1.5^x)
GCSE Quiz True

Using the points and y=ab^x, \, b>0.

 

(1, 9) gives 9=ab.

 

(4, 30.375) gives 30.375=ab^4.

 

Dividing gives

 

\begin{aligned} \cfrac{30.375}{9}&= \cfrac{ab^4}{ab} \\\\ 3.375 &= b^3\\\\ b&=1.5 \end{aligned}

 

Substituting into 9=ab gives

 

a=6.

Exponential function GCSE questions

1. Match the equations to the graphs.

 

Exponential Function GCSE 1 image 1 Exponential Function GCSE 1 image 2

Exponential Function GCSE 1 image 3 Exponential Function GCSE 1 image 4

 

Exponential Function GCSE 1 image 5

 

(4 marks)

Show answer

C

(1)

D

(1)

A

(1)

B

(1)

2. Plot the graph of y= \left(\, \cfrac{1}{4} \, \right)^x, for x values between -1 and 1, on the grid below.

 

Exponential Function GCSE 2 image 1

 

(3 marks)

Show answer

2 correct points plotted.

(1)

4 correct points plotted.

(1)

Points joined with smooth curve.

(1)

 

Exponential Function GCSE 2 image 2

3. An exponential function in the form f(x)=ab^x, \, b>0 , goes through the points (1, 4) and (3, 256). Find the equation of the exponential function f(x).

 

(4 marks)

Show answer

Forms equations 4=ab and 256=ab^3.

(1)

Process to divide equations or b^2=64.

(1)

b=8

(1)

a=0.5

(1)

We use exponential functions when generating geometric sequences.

The nth term of a geometric sequence is given by u_n=ar^{n-1}. This is an exponential function.

Here is the nth term of a geometric sequence u_n=4(2)^{n-1}.

u_1=4(2)^{0}=4

u_2=4(2)^{1}=8

u_3=4(2)^{2}=16

Step-by-step guide: Geometric sequences

We have also seen exponential functions when generating the amount in a bank account which is receiving compound interest.

The compound interest formula is A=P\left(1+\cfrac{r}{100}\right)^n.

Where,

  • A represents the final amount.
  • P represents the original principal amount.
  • r is the interest rate.
  • n represents the number of times the interest rate is applied.

Step-by-step guide: Compound interest

Learning checklist

You have now learned how to:

  • Recognise, sketch and interpret graphs of exponential functions y=k^x, for positive values of k
  • Recognise the graph of exponential functions
  • Be able to find missing values from exponential functions that satisfy given coordinates

Exponential functions in A level mathematics

Exponential functions are explored in much more depth in A level mathematics.

We learn about a special number e.

e is a very important mathematical constant just like \pi .

e=2.7182818… and is an irrational number whose decimals never recur or terminate.

e is also known as Euler’s number after the Swiss mathematician Leonhard Euler, but it was actually discovered by another Swiss mathematician called Jacob Bernoulli whilst studying compound interest.

y=e^x is known as “the exponential function”.

e has some very special properties.

The value of the derivative of y=e^x is e^x. This means the gradient of the graph of y=e^x at any value x is the same as the value of e^x.

It can also be approximated using the formula \left(1+\cfrac{1}{n} \, \right)^n .

If you use increasing large values for n, the approximation will get closer and closer to e.

The inverse function of exponential functions are called logarithms.

Logarithmic functions have different bases depending on the base of the original exponential function.

The inverse of an exponential function y=2^x would be y=\log_2 x. We say “log base 2 ”.

The inverse of the exponential function y=e^x isn’t written as y=\log_e x but y=\ln x. This is known as the natural logarithm instead of saying “log base e”.

Logarithms are used to solve exponential equations, they can find the time for a bank investment to reach a required value or the half-life of a radioactive compound.

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