Quadratics and Harder Graphs Worksheets, Questions and Revision

Quadratics and Harder Graphs Worksheets, Questions and Revision

GCSE 4 - 5GCSE 6 - 7KS3AQAEdexcelOCRWJECFoundationHigherAQA 2022Edexcel 2022OCR 2022WJEC 2022

Quadratic Graphs and Other Graphs

This topic includes graphs which are not straight  lines.

These include, quadratic graphs, cubic graphsreciprocal graphs and exponential graphs.

You will need to be able to identify and plot these graphs.

You need to be happy with the following topics:

Quadratic Graphs

Quadratic graphs have the general form

\textcolor{red}{a}x^2 + \textcolor{limegreen}{b}x+\textcolor{blue}{c}

These form a \bigcup or \bigcap shape, examples are shown below:

quadratic graphs

Note: \textcolor{limegreen}{b} and \textcolor{blue}{c} can be zero, as is the case with y=x^2

Level 4-5GCSEKS3AQAEdexcelOCRWJEC
Level 4-5 GCSE AQA Edexcel OCR WJEC
cubic graph

Cubic Graphs 

Cubic graphs have the general form

\textcolor{red}{a}x^3 + \textcolor{limegreen}{b}x^2+\textcolor{blue}{c}x +\textcolor{maroon}{d}

These form S shape in the middle.

Note: Sometimes this S can be fairly flat, e.g.

\textcolor{red}{2}x^3 + \textcolor{limegreen}{3}x^2 + \textcolor{blue}{x}

cubic graph
Level 4-5 GCSE AQA Edexcel OCR WJEC
Level 6-7 GCSE AQA Edexcel OCR WJEC
reciprocal graph

Reciprocal Graphs

Reciprocal graphs have the general form

y = \dfrac{\textcolor{red}{k}}{x}

e.g.,

y = \dfrac{\textcolor{red}{1}}{x}

reciprocal graph
Level 6-7 GCSE AQA Edexcel OCR WJEC
exponential graph

Exponential Graphs

Exponential graphs have the general form

y = \textcolor{blue}{k}^x

e.g.,

y = \textcolor{blue}{3}^x

exponential graph
Level 6-7 GCSE AQA Edexcel OCR WJEC
Level 4-5 GCSE AQA Edexcel OCR WJEC

Example: Plotting Quadratics 

Plot the following quadratic equation:

y=x^2-x-5

[2 marks]

First  draw a table of coordinates from x=-2 to x=3, then use the values to plot the graph between these values of x.

Step 1: Draw a table for the values of x between -2 and 3.

Step 2: Substitute our values of x into the equation to get the corresponding y values.

For example, when x=\textcolor{red}{-2}, we get

y=(\textcolor{red}{-2})^2-(\textcolor{red}{-2})-5=4+2-5= \textcolor{blue}{1}.

Step 3: Continue this process for all other values of x

plotting quadratic graphs table of values

Step 4: From the table we get coordinates to plot. e.g. (\textcolor{red}{-2}, \textcolor{blue}{1})

Once plotted, we join all the points with a smooth curvegiving the following graph.

plotting quadratic graphsLevel 4-5GCSEKS3AQAEdexcelOCRWJEC
Level 4-5 GCSE KS3 AQA Edexcel OCR WJEC

Example: Plotting Cubics

Using the equation y=x^3-2x^2, draw a table of coordinates from x=-1 to x=3. Use the values to plot the graph between these x values.

[3 marks]

Step 1: Draw a table of the coordinates for x from -1 to 3

Step 2: Substitute our values of x into the equation to get the corresponding y values.

For example, for x=\textcolor{red}{1}, we get

y=\textcolor{red}{1}^3-2(\textcolor{red}{1})^2=\textcolor{blue}{-1}.

Step 3: Continue this process for all other values of x

plotting cubic graphs table of values

Step 4: From the table we get coordinates to plot. e.g. (\textcolor{red}{1}, \textcolor{blue}{-1})

Once plotted, we join all the points with a smooth curve giving the following graph.

plotting cubic graphsLevel 4-5GCSEAQAEdexcelOCRWJEC

Example Questions

We will complete this table by substituting in the values of x to get the missing values of y. For example, when x=2,

 

y=(-4)^2+4(-4)-9=16-16-9=-9

 

Continuing this with the rest of the x values, we get the completed table below.

 

quadratics and harder graphs example 1 answer table

 

Then, plotting these coordinates on a pair of axes and joining them with a curve, we get the graph below.

 

quadratics and harder graphs example 1 answer graph

 

We will complete this table by substituting in the values of x to get the missing values of y. For example, when x=-2,

 

y=(-2)^3+3(-2)^2-4=-8+12-4=0

 

Continuing this with the rest of the x values, we get the completed table below.

 

quadratics and harder graphs example 2 answer table

 

Then, plotting these points on a pair of axes and joining them with a curve, we get the graph below.

 

quadratics and harder graphs example 2 answer graph

 

We will complete this table by substituting in the values of x to get the missing values of y. For example, when x=2,

 

y=(0.2)^2=0.04

 

Continuing this with the rest of the x values, we get the completed table below.

 

quadratics and harder graphs example 3 answer table

 

Then, plotting these points on a pair of axes (to the best of your ability – some of the y values are so small they’re going to end up practically on the x-axis) and joining them with a curve, we get the graph below.

quadratics and harder graphs example 3 answer graph

 

We draw this table by substituting the x values into the equation. For example, for x=1, we get

y=2^1=2.

quadratics and harder graphs example 4 answer table

Carrying this on with the rest of the numbers, we get the table above. Then, plotting these points and joining them with a curve, we get the graph to the right.

The exponential graph also has an asymptote along the x-axis. Its shape varies very little, except that when the base of the exponential (here, the function is 2^x so the base is 2) is a number between 0 and 1, the shape of the graph is a mirror image of this one. Specifically, a reflection in the y-axis.

 

quadratics and harder graphs example 4 answer graph

We draw this table by substituting the x values into the equation. For example, for x=2, we get

y=\dfrac{1}{2}=0.5.

 

quadratics and harder graphs example 5 answer table

 

Then, plotting these points on a pair of axes and joining them with a curve, we get the graph below.

 

quadratics and harder graphs example 5 answer graph

 

Related Topics

MME

Drawing Straight Line Graphs Worksheets, Questions and Revision

Level 1-3GCSEKS3
MME

Coordinates and Ratios Worksheets, Questions and Revision

Level 6-7GCSEKS3

Worksheet and Example Questions

Site Logo

(NEW) Plotting Quadratics and Harder Graphs Exam Style Questions

Level 4-5 Level 6-7 GCSE

Drill Questions

Site Logo

Drawing Quadratic Graphs

Level 4-5 GCSE
Site Logo

Plotting Quadratics and Harder Graphs

Level 4-5 GCSE
Site Logo

Plotting Harder graphs

Level 6-7 GCSE

You May Also Like...

GCSE Maths Revision Cards

Revise for your GCSE maths exam using the most comprehensive maths revision cards available. These GCSE Maths revision cards are relevant for all major exam boards including AQA, OCR, Edexcel and WJEC.

£8.99
View Product

GCSE Maths Revision Guide

The MME GCSE maths revision guide covers the entire GCSE maths course with easy to understand examples, explanations and plenty of exam style questions. We also provide a separate answer book to make checking your answers easier!

From: £14.99
View Product

Transition Maths Cards

The transition maths cards are a perfect way to cover the higher level topics from GCSE whilst being introduced to new A level maths topics to help you prepare for year 12. Your ideal guide to getting started with A level maths!

£8.99
View Product