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GCSE Maths Algebra

Types of Graphs

Types of Graphs

Here we will learn about types of graphs, including straight line graphs, quadratic graphs, cubic graphs, reciprocal graphs, exponential graphs and circle graphs.

There are also types of 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 are types of graphs?

Types of graphs include different types of straight and curved graphs. 

We need to be able to recognise and distinguish between the main types of graphs.

What are types of graphs?

What are types of graphs?

Straight line graphs

Straight line graphs are graphs of linear functions and are of the form:

y=mx+c

Where m is the gradient and c is the y -intercept (where the line crosses the y -axis). 

The graphs look like this:

Types of graphs image 1 1

Step by step guide: Straight line graphs

Quadratic graphs

Quadratic graphs are graphs of a quadratic function and can be recognised as they include an squared term.
E.g.
An x^2 term. 

The shape of a quadratic graph is a parabola. 
The graphs have one tuning point – a minimum point or a maximum point.  

Types of graphs image 2 1

Step by step guide: Quadratic graphs

Cubic graphs

Cubic graphs are graphs of a cubic function and can be recognised as they include a cubed term.
E.g.
An x^3 term. 

The graphs often have two turning points – a minimum point and a maximum point. 

Types of graphs image 3

Step by step guide: Cubic graphs

Exponential graphs

Exponential graphs are graphs of an exponential function and can be recognised as they include a k^x term where k is the base and x is the exponent (power). 
The graphs can be a growth curve when k is greater than 1 or a decay curve when k is less than 1 .

Types of graphs image 4 1

More complex exponential curves are of the form:

y=ab^x

Step by step guide: Exponential graphs

Reciprocal graphs

Reciproccal graphs are graphs of a reciprocal function and can be recognised as they include a \frac{1}{x} term. 
The graphs can be recognised as they have two separate parts, often in different quadrants of the coordinate grid.

A simple reciprocal graph
 
y=\frac{1}{x}

A more complex reciprocal graph
 
y=\frac{1}{x} +x

Types of graphs image 5 1

Step by step guide: Reciprocal graphs

Circle graphs

Circle graphs at GCSE are graphs of a circle with centre (0,0)
They are of the form:

x^2+y^2=r^2
Where r is the radius of the circle.

E.g.
This circle graph has:
 
centre (0,0)
 
radius 3
 
Its equation is:
 
x^2+y^2=3^2
 
Which can be simplified to:
 
x^2+y^2=9

                    Types of graphs image 6 1

Step by step guide: Circle graphs

How to recognise types of graphs

In order to recognise and discriminate the types of graphs:

  1. Identify linear or quadratic.
  2. Identify if there is a circle graph.
  3. Identify other curves by looking at the features such as growth, or vertices.
  4. Identify all the graphs clearly.

Types of graphs worksheet

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

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Types of graphs worksheet

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

COMING SOON

Recognising types of graphs examples

Example 1: recognise the types of graphs

Match the graph with its equation

Types of graphs example 1 1 Graph A

Types of graphs example 1 1 Graph B

Types of graphs example 1 (2) 1 Graph C

Types of graphs example 1 (2) 1Graph D

Equation 1
 
y=2^x
 
Equation 3
 
x^2+y^2=4

Equation 2
 
y=x^2+x-2
 
Equation 4
 
y=x^3-x-2

  1. Identify linear or quadratic.

There is no straight line graph, so there is no linear function.

There is a parabola graph, so this is the graph of a quadratic function which has an x^2 term.

Graph C is Equation 2

2Identify if there is circle graph.

There is a circle graph.  It has centre (0,0) and radius 2 ; so its equation would be:

x^2+y^2=2^2

Graph D is Equation 3

3Identify other curves by looking at the features such as growth, or vertices.

Graph A has 2 vertices; it is very likely to be a cubic function. 
Equation 4 is a cubic function with a y -intercept at −2 which is at the end of the equation for when x = 0 .

Graph B is a growth curve so its equation will have a term with x as an exponent (power); equation 1 has a term with x as an exponent.

4Identify all the graphs clearly.

Types of graphs example 1 step 4 1 Graph A = Equation 4

Types of graphs example 1 step 4 1 Graph B = Equation 1

Types of graphs example 1 step 4 (2) 1 Graph C = Equation 2

Types of graphs example 1 step 4 (2) 1 Graph D = Equation 3

Example 2: recognise the types of graphs

Match the graph with its equation

Types of graphs example 2 1 Graph A

Types of graphs example 2 1 Graph B

Types of graphs example 2 1 Graph C

Types of graphs example 2 1 Graph D

Equation 1
 
x+y=5
 
Equation 3
 
x^2+y^2=25

Equation 2
 
y=\frac{5}{x}
 
Equation 4
 
y=0.5^x

There is a straight line graph so this is the graph of a linear function which has no visible powers.


There is no parabola graph so there is no quadratic function.


Graph B is Equation 1

There is a circle graph.  It has centre (0,0) so its equation would be:


x^2+y^2=r^2


Graph A is Equation 3

Graph C is not a growth curve but it is a decay curve so its equation will have a term with x as an exponent (power).  Equation 4 has a term with x as an exponent.


Graph D is a curve which has 2 sections in different quadrants of the coordinate grid.  It is likely to be a reciprocal graph with a \frac{1}{x} term.  Equation 2 has a term with with x as a denominator of a fraction.

Types of graphs example 2 step 4 Graph A = Equation 3

Types of graphs example 2 step 4 Graph B = Equation 1

Types of graphs example 2 step 4 (2) Graph C = Equation 4

Types of graphs example 2 step 4 (2) Graph D = Equation 2

How to use types of graphs

In order to use different types of graph to solve an equation:

  1. Add a line to the coordinate grid.
  2. See where the line crosses the curve.
  3. Draw a straight vertical line from the curve to the x -axis.
  4. Read off the value on the x -axis.

Using types of graphs examples

Example 3: using types of graphs

Use the graph of y=1.5^x to find an approximate solution to the equation 1.5^x=7

Types of graphs example 5 1

Since the y in the equation of the curve has been replaced by the 7 , add the horizontal line y=7 to the coordinate grid.


Make sure you use a ruler and be as accurate as you can.


Types of graphs example 5 step1 1

The line y=7 crosses the curve at one point.


Types of graphs example 5 step 2 1

Continue to use a ruler and try to be as accurate as you can.


Types of graphs example 5 step 3 1

Be careful with the scale on x -axis.  The solution to the equation is only approximate, but try to be as accurate as you can.


x=4.8

Example 4: using types of graphs

Use the graph of y=\frac{1}{x} to find an approximate solution to the equation \frac{1}{x}=x+2


Types of graphs example 6 1

Since the y in the equation of the curve has been replaced by the x + 2 , add the horizontal line y = x + 2 to the coordinate grid.


Make sure you use a ruler and be as accurate as you can.


Types of graphs example 6 step 1 1

The line y = x + 2 crosses the curve at two points.


Types of graphs example 6 step 2 1

Continue to use a ruler and try to be as accurate as you can.


Types of graphs example 6 step 3 1

Be careful with the scale on x -axis.  The solution to the equation is only approximate, but try to be as accurate as you can.


x=0.4 \quad \text{or} \quad x=-2.4

How to plot types of graphs

In order to plot different types of graphs:

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

Plotting types of graphs examples

Example 5: Plot types of graphs

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

y=x^3-2x+3

Substitute the values of x into the equation.  Write the values of y in the table.


\begin{aligned} &x \quad \quad -2 \quad \quad -1 \quad \quad 0\quad \quad 1 \quad \quad 2 \\ &y \quad \quad -1 \quad \quad \quad 4 \quad \quad 3 \quad \quad 2 \quad \quad 7 \end{aligned}

It is best practice to use a pencil and plot the coordinates using small crosses.


The coordinates would be (-2,-1), (-1,4) and so on.  Note that the y -interceptis +3 which is also at the end of the equation for when x = 0.


Types of graphs example 5 step 2 2

Use a pencil and turn the paper around if it makes it easier for you.

Example 6: Plot types of graphs

Draw the curve for 1\leq{x}\leq18 :

y=\frac{18}{x}

Substitute the values of x into the equation.  Write the values of y in the table.


\begin{aligned} &x \quad \quad 1 \quad \quad 2 \quad \quad 3 \quad \quad 6 \quad \quad 9 \quad \quad 18\\ &y \quad \quad 18 \quad \;\;9 \quad \quad 6 \quad \quad 3 \quad \quad 2 \quad \quad 1 \end{aligned}

It is best practice to use a pencil and plot the coordinates using small crosses.


The coordinates would be (1,18), (2,9) and so on.


Types of graphs example 6 step 2 2

Use a pencil and turn the paper around if it makes it easier for you.


Types of graphs example 6 step 3 2

Common misconceptions

  • Smooth curve

Points should be joined with a smooth curve – NOT line segments like the example below. 

Types of graphs Common Misconceptions 1

  • Points on the curve

All points should be on the curve. If there is a point that does not join up with the other points when you draw a smooth curve – go back and check the coordinate.  In the example below there is a point that does not lie on the smooth curve so needs checking.

Types of graphs Common Misconceptions 2 1

Practice types of graphs questions

1. Identify the correct equation for this graph:

 

Types of graphs Practice Question 1 1

y=5^x
GCSE Quiz True

y=x^3+5x
GCSE Quiz False

y=\frac{5}{x}
GCSE Quiz False

x^2+y^2=25
GCSE Quiz False

The graph shows a growth curve. We need to look for an equation which has a term like k^x where x is the exponent (power).

2. Identify the correct equation for this graph:

 

Types of graphs Practice Question 2 1

y=\frac{1}{x+1}
GCSE Quiz False

y=x^3-5x+1
GCSE Quiz True

x^2+y^2=10
GCSE Quiz False

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

The graph shows a curve with 2 vertices – a minimum point and a maximum point.

 

We need to look for an equation which has a x^3 term.

 

Also the y -intercept is positive, and the equation ends +1 for when x=0.

3. Identify the correct graph for this equation:

 

x^2+y^2=9

Types of graphs Practice Question 3 Correct Answer 1 1

GCSE Quiz True

Types of graphs Practice Question 3 Correct Answer 2 1

GCSE Quiz False

Types of graphs Practice Question 3 Correct Answer 3 1

GCSE Quiz False

Types of graphs Practice Question 3 Correct Answer 4 1

GCSE Quiz False

The equation is of the form:

 

x^2+y^2=r^2

 

So we are looking for a circle with centre (0,0).

 

\begin{aligned} r^2 &= 9 \\\\ r &= \sqrt{9} = 3 \end{aligned}

 

The radius of the circle is 3 .

 

So the circle will pass through 3 on the x-axis and the y-axis.

4. Identify the correct graph for this equation:

 

y=-x^3+2x+3

Types of graphs Practice Question 4 - 3 1

GCSE Quiz False

Types of graphs Practice Question 4 - 2 1

GCSE Quiz False

Types of graphs Practice Question 4 1

GCSE Quiz True

The equation is a cubic with a negative x^3 term.

 

So we look for a cubic graph which shows as x increases so does y but negatively.

 

So this is the approximate shape:

 

Types of graphs Practice Question 4 explanation 1

 

The equation also ends with +3 for when x=0.

 

So this tells us the y -intercept is also +3

5. Use the graph of y=4^x to find an approximate solution to:

 

4^x=3

 

Types of graphs Practice Question 5 1

1.8
GCSE Quiz False

0.8
GCSE Quiz True

0.4
GCSE Quiz False

1.4
GCSE Quiz False

Draw a horizontal line at y=3 and see where it crosses the line.

 

Then draw a vertical line down and read the value of the x-axis.

 

Types of graphs Practice Question 5 explanation 1

6. Use the graph of y=\frac{1}{x-2} to find an approximate solution to:

 

\frac{1}{x-2}=1.4

 

Types of graphs Practice Question 6 1

2.7
GCSE Quiz True

3.7
GCSE Quiz False

3.2
GCSE Quiz False

2.2
GCSE Quiz False

Draw a horizontal line at y=1.4 and see where it crosses the line.

 

Then draw a vertical line down and read the value of the x-axis.

 

Types of graphs Practice Question 6 2

Types of graphs GCSE questions

1.  On the grid, sketch the curve with the equation

 

y=3^x

 

Give the coordinates of any points of intersection with the axes.

 

Types of graphs GCSE question 1 1

 

(2 marks)

Show answer

Types of graphs GCSE question 1 (2) 1

 

for drawing a growth curve

(1)

for the y -intercept (0,1)

(1)

2. Here are three graphs:

Types of graphs GCSE question 2 1

 

Here are three equations:

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

 

Match each graph to the correct equation

 

Graph A and y = ………………

Graph B and y = ………………

Graph C and y = ……………….

 

(3 marks)

Show answer

Graph A and y=x^3

(1)

Graph B and y=x^{2}+2

(1)

Graph C and y=\frac{1}{x}

(1)

3. (a)  Complete the table of values for y=x^{3}-6x+1

 

\begin{aligned} &x \quad \quad -3 \quad \quad -2 \quad \quad -1 \quad \quad 0 \quad \quad 1 \quad \quad 2 \quad \quad 3\\ &y \quad \quad -0 \quad \quad \quad \quad \quad \quad \;6 \quad \quad \quad \;\quad \quad \quad -3 \quad \quad 10 \end{aligned}

 

(b)      On the grid, draw the graph of y=x^{3}-6x+1 for -3\leq{x}\leq3

 

Types of graphs GCSE question 3b 1

 

(c) Use the graph to find an approximate solution for 

 

x^{3}-6x+1=7

(5 marks)

Show answer

(a)

 

Correct Values are: 5, 1 and -4

 

For 1 correct y -values

(1)

For all correct y -values

(1)

 

(b)

 

Types of graphs GCSE question 4b 1

 

for plotting the points correctly

(1)

for the smooth curve

(1)

 

(c)

 

x=2.8

 

approximate solution

(1)

Learning checklist

You have now learned how to:

  • Recognise and sketch different types of curved graphs
  • Plot different types of curved graphs
  • Use and interpret different types of curved graphs

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