Quadratics and Harder Graphs Worksheets & Revision | MME

Quadratics and Harder Graphs Worksheets, Questions and Revision

Level 6-7

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

KS3 Level 4-5
Level 4-5

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
cubic graph
Level 4-5
Level 6-7

Reciprocal Graphs

Reciprocal graphs have the general form

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

e.g.,

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

reciprocal graph
reciprocal graph
Level 6-7

Exponential Graphs

Exponential graphs have the general form

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

e.g.,

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

exponential graph
exponential graph
Level 6-7
Level 4-5

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 graphs
KS3 Level 4-5
KS3 Level 4-5

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 graphs
Level 4-5

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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

 

Worksheets and Exam Questions

MME

(NEW) Plotting Quadratics and Harder Graphs Exam Style Questions

Level 5-6 New Official MME

Drill Questions

MME

Drawing Quadratic Graphs

MME

Plotting Quadratics and Harder Graphs

MME

Plotting Harder graphs