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Substitution Coordinates maths Straight line graphs Powers and rootsThis topic is relevant for:

Here we will learn about **cubic graphs**, including recognising and sketching cubic graphs. We will also look at plotting and interpreting cubic graphs.

There are also cubic graphs worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if youβre still stuck.

A** cubic graph** is a graphical representation of a cubic function.

A **cubic **is a polynomial which has an ^{3}

These graphs have:

- a
**point of inflection**where the curvature of the graph changes between concave and convex - either
**zero or two turning points**

A cubic graph with two turning points can touch or cross the x axis between one and three times.

1 Inflection Point
0 Turning Points E.g. y=x^3 | 1 Inflection Point
2 Turning Points E.g. y=5x^3+x^2-3x |
---|---|

E.g.

In the graph below the coefficient of the x^{3} term is positive so the graph increases

y=x^{3}-2x^{2}-x+2Also, the y -intercept of the curve is +2 and the equation ends with +2 for when x=0.

E.g.

In the graph below the coefficient of the x^3 term is negative so the graph decreases

y=-x^{3}+2x^{2}+x-2Also, the y -intercept of the curve is -2 and the equation ends with -2 for when x=0.

In order to recognise a cubic graph:

**Identify linear or quadratic or any other functions.****Identify the cubic function checking if the**x term is positive or negative.^{3}**Identify your final answer.**

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

DOWNLOAD FREEGet your free cubic graph worksheet of 20+ types of graphs questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE**Cubic 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:

Identify the correct graph for the equation:

\[y=x^3+2x^2+7x+4\]

**Identify linear or quadratic or any other functions**.

Graph

Graph

2**Identify the cubic function checking if the **^{3}**term is positive or negative**.

Graph

The given equation has a positive ^{3}

For Graph

Graph

3**Identify your final answer**.

Graph

Identify the correct graph for the equation:

\[y=-x^3+2x^2+4\]

**Identify linear or quadratic or any other functions**.

Graph

Graph

**Identify the cubic function checking if the ** x^3 **term is positive or negative**.

Graph

The given equation has a negative ^{3}

Graph

Graph

**Identify your final answer**.

Graph

In order to plot a cubic graph:

**Complete the table of values.****Plot the coordinates.****Draw a smooth curve through the points.**

\[y=x^{3}-3x-1\]

**Complete the table of values**.

**Plot the coordinates**.

Draw an

The coordinates would be

**Draw a smooth curve through the points**.

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

Note: the

\[y=x^3-6x+1\]

**Complete the table of values**.

**Plot the coordinates**.

Draw an

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

**Draw a smooth curve through the points**.

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

Note: the

In order to use a cubic graph to solve an equation:

**Find the given value on the**y -axis.**Draw a straight horizontal line across the curve.****Draw a straight vertical line from the curve to the**x -axis.**Read off the value on the**x -axis.

Use the graph of y = x^3 - 5x + 1 to find an approximate solution of the following equation:

\[x^3-5x+1=6\]

**Find the given value on the **** y-axis**.

The value of

We need to look on the

**Draw a straight horizontal line across the curve**.

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

**Draw a straight vertical line from the curve to the**

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

**Read off the value on the **

Be careful with the scale on

You could substitute the

Use the graph of y = x^3 - 7x - 3 to find approximate solutions of the following equation:

\[x^3-7x-3=2\]

**Find the given value on the **** y-axis**.

The value of

**Draw a straight horizontal line across the curve**.

Make sure you use a ruler and be as accurate as you can. The horizontal line intersects the curve in

**Draw a straight vertical line from the curve to the**

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

**Read off the value on the **

Be careful with the scale on

**Scale on the**x andy axis

Cubic graphs often have different scales on the

**The curve must be smooth**

Cubic graphs need to be drawn with a smooth curve. Avoid straight line segments.

Incorrect:

Correct:

**Cubic graphs have at the most only**2 turning points

Make sure that your cubic graph has only one minimum point and one maximum point. Check that you have the correct values and that you have plotted them accurately.

**You can use the**y -intercept to help distinguish between cubic curves

The

\[y=x^3+x^2+4\]

1. Identify the correct graph for the below equation:

y=x^3-3x+2

Graph A:

GraphΒ B:

Graph C:

Graph D:

Graph A

Graph B

Graph C

Graph D

Graph A is a growth curve so is an exponential function

Graph B has 2 turning points so could be the graph of a cubic function.

Graph C is a parabola so is a quadratic function.

Graph D is a hyperbola so is a reciprocal function.

Graph B is the only curve which could be a cubic function.

Also the y -intercept is positive on the curve and the equation.

2. Identify the correct graph for the equation:

y=-x^3+2x-3

Graph A:

Graph B:

Graph C:

Graph D:

Graph A

Graph B

Graph C

Graph D

All graphs have two turning points so all the graphs could be graphs of cubic functions.

Graph B and Graph C show that as x increases so does y so they have a positive x^3 term. So the correct graph is not one of these.

Graph A and Graph D show that as x increases so does y but negatively so they have a negative x^3 term.

But the y -intercept on Graph A is negative, and the given equation ends with -3 so this is the correct graph.

3. Identify the correct graph for the equation:

y=x^{3}-2x+3

Graph A:

Graph B

&nbso;

Graph C:

Graph D:

Graph A

Graph B

Graph C

Graph D

A correct table of values would be:

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

These values give coordinates such as (-2,-1), (-1,4) and so on.

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=x^3-4x+5

Graph A:

Graph B:

Graph C:

Graph D :

Graph A

Graph B

Graph C

Graph D

A correct table of values would be:

\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 -10 \quad \quad \;\,5 \quad \quad\quad 8 \quad \quad 5 \quad \quad 2 \quad \quad 5 \quad \quad 20 \end{aligned}

These values give coordinates such as (-3,-10), (-2,5) and so on.

The coordinates are plotted on the grid.

A smooth curve should be drawn through the points.

5. Use the graph of y=6+3x-x^3 to solve:

6+3x-x^3=2

2.2

2.5

2.9

1.9

6. Use the graph of y=x^3-3x+7 to solve:

x^3-3x+7=4

-2.1

2.1

-1.9

1.9

1.Β Match the correct equation to the graph:

y=x^2+3x+2 \quad \quad y=3x+2 \quad \quad y=x^3-5x+2 \quad \quad y=\frac{3}{x}

**(1 mark)**

Show answer

y=x^3-5x+2

**(1)**

2.

(a) Complete the table of values for y=x^{3}-8x+3

(b) On the grid, draw the graph of y=x^{3}-8x+3 for

-3\leq{x}\leq3

**(4 marks)**

Show answer

(a)

Correct values are: 10, 3 and -5.

For 1 correct y -values.

**(1)**

For all correct y -values.

**(1)**

(b)

For plotting the points correctly.

**(1)**

For the smooth curve.

**(1)**

3.

(a) Complete the table of values for y=x^{3}-6x+2

(b) On the grid, draw the graph of y=x^{3}-6x+2 for

-3\leq{x}\leq3

(c) Use the graph to solve the equation:

x^3-6x+2=8

**(5 marks)**

Show answer

(a)

Correct values are: 6, 2 and -3.

For 1 correct y -value.

**(1)**

For all correct y -values.

**(1)**

(b)

For plotting the points correctly.

**(1)**

For the smooth curve.

**(1)**

(c)

For answer between 2.7 and 2.9.

**(1)**

You have now learned how to:

- Recognise graphs of cubic functions
- Draw graphs of cubic functions
- Interpret graphs of cubic functions

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