GCSE Tutoring Programme

Our chosen students improved 1.19 of a grade on average - 0.45 more than those who didn't have the tutoring.

Teacher-trusted tutoring
GCSE Maths Algebra Types Of Graphs

Exponential Graph

Exponential Graph

Here we will learn about exponential graphs, including recognising, sketching, plotting and interpreting exponential graphs.

There are also exponential graphs 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 graph?

An exponential graph is a representation of an exponential function of the form

\[y=k^x\]

Where x and y are variables and k is a constant (a numerical value).  
x is the exponent and k is the base.

The y-intercept of an exponential curve (at x = 0 ) is 1 since any non-zero number raised to the power 0 is 1.

The x-axis is an asymptote to the curve.
The curve gets very close to the horizontal asymptote but does not touch it. This is because y β‰  0.

The graph of an exponential function can represent either exponential growth or exponential decay:

  • When k is greater than 1 it is a growth curve. As x increases, so does y.
    It can be used to represent population growth or compound interest. 

E.g.

\[y=2^x\]
Exponential Graph image 1 1

  • When k is less than 1 it is a decay curve. As x increases, y decreases.
    It can be used to represent radioactive decay.

E.g.

\[y=(\frac{1}{2})^x\]
Exponential Graph image 2 1

There are more complex exponential functions of the form:

\[y=ab^x\]

The graphs look similar to the ones above, they have an exponent x, base b and the y-intercept is a.

E.g.

\[y=4\times 3^x\]
Exponential Graph image 3 1

What is an exponential graph?

What is an exponential graph?

Growth and decay problems

Exponential graphs can represent two types of problems, growth and decay. They involve real-life situations.


Growth problems are when the amount increases over time. An example is how a population increases over time.

Decay problems are when the amount decreases over time. An example is  the amount of radiation from a radioactive substance over time.

How to recognise an exponential graph

In order to recognise an exponential graph:

  1. Identify linear or quadratic or any other functions
  2. Identify the exponential function
  3. Identify your final answer

Types of graphs worksheet (includes exponential graphs)

Types of graphs worksheet (includes exponential graphs)

Types of graphs worksheet (includes exponential graphs)

Get your free exponential graph worksheet of 20+ types of graphs questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE
x
Types of graphs worksheet (includes exponential graphs)

Types of graphs worksheet (includes exponential graphs)

Types of graphs worksheet (includes exponential graphs)

Get your free exponential graph worksheet of 20+ types of graphs questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE

Related lessons on types of graphs

Exponential graph is part of our series of lessons to support revision on types of graphs. You may find it helpful to start with the main types of graphs lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

Recognising an exponential graph examples

Example 1: recognise exponential graphs

Which is the correct equation for the graph?

Exponential Graph example 1 step 1 1

\[y=x^2+3 \quad \quad \quad y=3x+1 \quad \quad \quad y=\frac{3}{x} \quad \quad \quad y=3^x\]

  1. Identify linear or quadratic or any other functions

y=x^2+3 is a quadratic function; its graph would be a parabola.

y=3x+1 is a linear function; its graph would be a straight line.

y=\frac{3}{x} is a reciprocal function; its graph would be a hyperbola.

2Identify the exponential function

y=3^x is an exponential function.

It has x as the exponent and the base is 3, which is greater than 1.

So the graph would be a growth curve.

3Identify your final answer

Exponential Graph example 1 step 3 1

The correct equation for the graph is y=3^x

Example 2: recognise exponential graphs

Which is the correct equation for the graph?

Exponential Graph example 2 step 1 1

\[y=1-3x\quad\quad\quad y=x^2-3 \quad\quad\quad y=(\frac{1}{3})^x \quad\quad\quad y=x^{3} \]

Identify linear or quadratic or any other functions

Identify the exponential function

Identify your final answer

How to plot an exponential graph

In order to plot an exponential graph:

  1. Complete the table of values
  2. Plot the coordinates
  3. Draw a smooth curve through the points

Explain how to plot an exponential graph

Explain how to plot an exponential graph

Plotting an exponential graph examples

Example 3: plot exponential graphs

Draw the curve of the equation for -1\leq{x}\leq3

y=2^x

Complete the table of values

Plot the coordinates

Draw a smooth curve through the points

Example 4: plot exponential graphs

Draw the curve of the equation for -2\leq{x}\leq2

y=(\frac{1}{2})^x

Complete the table of values

Plot the coordinates

Draw a smooth curve through the points

How to find the equation of an exponential graph

In order to find the equation of an exponential graph:

  1. Substitute the pairs of values into the given equation
  2. Solve the two simultaneous equations
  3. Write down the equation of the exponential function

Finding the equation of an exponential graph example

Example 5: finding the equation of an exponential graph

The sketch shows a curve with equation y = ab^x where a and b are constants and b > 0.

Find the equation of the curve.


Exponential Graph example 5 step 1 1

Substitute the pairs of values into the given equation

Solve the two simultaneous equations

Write down the equation of the exponential function

Example 6: finding the equation of an exponential graph

The sketch shows a curve with equation y = ab^x where a and b are constants and b > 0.

Find the equation of the curve.


Exponential Graph example 6 1

Substitute the pairs of values into the given equation

Solve the two simultaneous equations

Write down the equation of the exponential function

Common misconceptions

  • Curves should be smooth

Do NOT join up the plotted points with straight lines like this:

Exponential Graph common misconceptions 1 1


The points should be joined by a single, smooth curve.

  • All points should be on the curve

If there is a point that does not join up with the other points when a smooth curve is drawn, this point is incorrect.

Exponential Graph common misconceptions 2 2

Practice exponential graph questions

1. Identify the correct equation for the graph:

 

Exponential Graph Practice Questions 1 1

y=6^x
GCSE Quiz True

y=6x+1
GCSE Quiz False

y=x^2 +1
GCSE Quiz False

y=\frac{6}{x}
GCSE Quiz False

The curve is a growth curve; its equation will be an exponential function.

 

y=6^x is an exponential function.

 

y=6x+1 is a linear function; its graph would be a straight line.

 

y=x^2+1 is a quadratic function; its graph would be a parabola.

 

y=\frac{6}{x} is a reciprocal function; its graph would be a hyperbola.

2. Identify the correct equation for the graph:

 

Exponential Graph Practice 2 1

y=1-10x
GCSE Quiz False

y=0.1^x
GCSE Quiz True

y=1-x^2
GCSE Quiz False

y=x^3+1
GCSE Quiz False

The curve is a decay curve; its equation will be an exponential function.

 

y=0.1^x is an exponential function.

 

y=1-10x is a linear function; its graph would be a straight line.

 

y=x^2+1 is a quadratic function; its graph would be a parabola.

 

y=x^3+1 is a cubic function.

Identify the correct equation for the graph:

 

y=10^x

Exponential Graph Practice 3 A

GCSE Quiz True

Exponential Graph Practice 3B

GCSE Quiz False

Exponential Graph Practice 3C

GCSE Quiz False

Exponential Graph Practice 3D

GCSE Quiz False

A correct table of values would be:

 

0 1 2
1 10 100

 

These x values and y values give coordinates such as (0, 1), (1, 10) and (2,100).

 

The coordinates are plotted on the grid.

 

A smooth curve should be drawn through the points.

4. Identify the correct graph for the equation:

 

y=(\frac{1}{5})^x

Exponential Graph practice questions 4a

GCSE Quiz False

Exponential Graph practice questions 4B

GCSE Quiz False

Exponential Graph practice questions 4C

GCSE Quiz False

Exponential Graph practice questions 4D

GCSE Quiz True

A correct table of values would be:

 

-2 -1 0 1
25 5 1 0.2

 

These x values and y values give coordinates such as (-2, 25), (-1, 5) and so on.

 

The coordinates are plotted on the grid.

 

A smooth curve should be drawn through the points.

5. Find the equation of the curve in the form

 

y=ab^x

 

where a and b are constants and b > 0.

 

Exponential Graph practice questions 5 1

y=3\times 4^x
GCSE Quiz False

y=4\times 3^x
GCSE Quiz False

y=2\times 4^x
GCSE Quiz False

y=4\times 2^x
GCSE Quiz True

Substituting the coordinates into the given equation gives

 

4=ab^0=a

 

Now we need to find the base b

 

32 = ab^3
32= 4\times b^3
8 = b^3
b = \sqrt[3]{8}
b = 2

 

So the equation is:

 

y=4\times 2^x

6. Find the equation of the curve in the form

 

y=ab^x

 

where a and b are constants and b > 0.

 

Exponential Graph practice questions 5 2

y=7\times 3^x
GCSE Quiz True

y=4\times 3^x
GCSE Quiz False

y=2\times 4^x
GCSE Quiz False

y=3\times 4^x
GCSE Quiz False

Substituting the coordinates into the given equation gives

 

21=ab^1=ab

 

189=ab^3

 

Dividing the two equations

 

\frac{189}{21} = \frac{ab^3}{ab}
9 = b^2
b = \sqrt{9}
b = 3

 

We can substitute b = 3 into one of the equations to find a.

 

21 = ab
21 = a\times 7
a = 7

 

So the equation is:

 

y=7\times 3^x

Exponential graph GCSE questions

1.Β  Match the correct graph to the equation.

 

y=8^{x}

 

Exponential Graph GCSE question 1

 

(1 Mark)

Show answer

Graph C

(1)

2.Β  (a) Complete the table of values for y=4^x

 

\begin{aligned} &x \quad \quad -1 \quad \quad 0 \quad \quad 1 \quad \quad 2 \\ &y \end{aligned}

 

(b) On the grid, draw the graph of y=4^{x} for -1\leq{x}\leq2

 

Exponential Graph GCSE 2B 1

 

(4 Marks)

Show answer

(a)

 

Correct values are: 0.25, 1, 4 and 16

 

For 1 correct y -value

(1)

 

For all correct y -values

(1)

 

(b)

 

for plotting the 4 points correctly

(1)

 

for the smooth curve

(1)

 

Exponential Graph GCSE 2B(2) 1

3. Here is a sketch of part of the graph y=pq^{x} where q\gt0

 

Exponential Graph GCSE question 3 1

 

The points (0,3), (2,k) and (4,1875) are all on the graph y=pq^{x}

 

Find the value of k.

(4 Marks)

Show answer

3=pq^0 = p .

 

for p = 3

(1)

 

1875=pq^4 = 3\times q^4 .

(1)

 

q=\sqrt[4]{625}=5 .

 

for q=5

(1)

 

k=3\times 5^2=75 .

 

for the correct value of k

(1)

Learning checklist

You have now learned how to:

  • Recognise graphs of exponential functions
  • Draw graphs of exponential functions
  • Interpret graphs of exponential functions

Still stuck?

Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors.

GCSE Benefits

Find out more about our GCSE maths tuition programme.