Quadratics and Harder Graphs

GCSEKS3Level 4-5Level 6-7AQAEdexcelOCR

Quadratics and Harder Graphs Revision

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:

Level 4-5GCSEAQAEdexcelOCR

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 4-5GCSEKS3AQAEdexcelOCR
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 4-5GCSEAQAEdexcelOCR
Level 4-5GCSEAQAEdexcelOCR

Interpreting Roots and Intercepts

The roots of a quadratic are the values of x where the quadratic equals 0, so if 

y= ax^2+bx+c

then the roots of ax^2+bx+c are the values of x when y=0. When the quadratic is graphed, the roots are the points on the graph where it crosses the x-axis. These points are called the x-intercepts.

So if x_1 and x_2 are roots of the quadratic, then the point (x_1, 0) and (x_2, 0) are on the graph. 

In this example, the roots of the quadratic can be interpreted graphically as the points that the curve crosses the x-axis. Giving us

(-3,0) and (3,0)

So the roots are x=3 and x = -3

The y-intercept is where the curve crosses the x-axis. In this case it is at the point

(0,-3)

So the y-intercept is y=-3

The graphical interpretation of the x and y-intercepts can also be done for other types of graphs such as cubic graphs.

Level 4-5GCSEAQAEdexcelOCR
Level 4-5GCSEAQAEdexcelOCR
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 4-5GCSEAQAEdexcelOCR
Level 6-7GCSEAQAEdexcelOCR
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 6-7GCSEAQAEdexcelOCR
Level 4-5GCSEAQAEdexcelOCR

Example 1: 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
Level 4-5GCSEKS3AQAEdexcelOCR
Level 4-5GCSEKS3AQAEdexcelOCR

Example 2: 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 4-5GCSEAQAEdexcelOCR

Example 3: Plotting Reciprocals

Using the equation y= \dfrac{1}{x} draw a table of coordinates from x = -2.5 to x=2.5. Use the values to plot the graph between these x values.

[3 marks]

Step 1: Draw a table of the coordinates for x from -2.5 to 2.5

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

For example, for x = -2, we get

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

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

Step 4: From the table we get coordinates to plot. e.g. (-2, -0.5)

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

Level 4-5GCSEAQAEdexcelOCR

Example 4: Plotting Exponentials

Using the equation y= 2^x 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 = -1, we get

y= 2^-1 =0.5

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

Step 4: From the table we get coordinates to plot. e.g. (-1, 0.5)

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

Level 6-7GCSEAQAEdexcelOCR

Quadratics and Harder Graphs 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

 

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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

 

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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

 

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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

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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

 

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Quadratics and Harder Graphs Worksheet and Example Questions

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(NEW) Plotting Quadratics and Harder Graphs Exam Style Questions

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Quadratics and Harder Graphs Drill Questions

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Drawing Quadratic Graphs

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Plotting Quadratics and Harder Graphs

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Plotting Harder graphs

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