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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.

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).

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

In general,

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

**Note that ** b ** will always be positive.**

In order to draw exponential graphs:

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

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

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

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

**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.

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

3**Join the points with a smooth curve.**

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

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

We have x values from -3 to 2.

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

P**lot the points on an ** \textbf{xy} ** coordinate plane**.

**Join the points with a smooth curve.**

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

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

We have x values from -2 to 2.

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

P**lot the points on an ** \textbf{xy} ** coordinate plane**.

**Join the points with a smooth curve.**

In order to find exponential functions:

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

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

**Find two points that lie on the graph**.

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

**Form two equations in the form** \bf{y=ab^x} **.**

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

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

**Solve the equations simultaneously.**

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

b=2.5 .

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

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

**Find two points that lie on the graph. **

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

**Form two equations in the form ** \bf{y=ab^x} **.**

(1, 4.5) gives us 4.5=ab.

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

**Solve the equations simultaneously.**

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).

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

**Find two points that lie on the graph. **

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

**Form two equations in the form ** \bf{y=ab^x} **.**

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

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

**Solve the equations simultaneously.**

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).

**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.

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

y=2^x has the table of values,

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

A

B

C

D

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

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

A

B

C

D

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

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

y=5^x

y=x^5

y=25^x

y=12.5x

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)

y=3(4^x)

y=3(2^x)

y=4^x

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)

y=1.5(6^x)

y=6(2.5^x)

y=6(1.5^x)

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.

1. Match the equations to the graphs.

**(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.

**(3 marks)**

Show answer

2 correct points plotted.

**(1)**

4 correct points plotted.

**(1)**

Points joined with smooth curve.

**(1)**

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

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 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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