# Quadratic, Cubic and Harder Graphs

GCSELevel 4-5Level 6-7Edexcel iGCSE

## Quadratic, Cubic and Harder Graphs

Graphs can take the shape of many forms; quadratic, cubic, and reciprocal. It is important you know how to identify and plot each of these types.

## Quadratic Graphs – General Shape

$y=\textcolor{#00d865}{a}x^2+\textcolor{#00d865}{b}x+\textcolor{#00d865}{c}$

Quadratic graphs can either be positive ($a>0$) or negative ($a<0$)

Note:

• $\textcolor{#00d865}{b}$ and $\textcolor{#00d865}{c}$ can be zero. In this case, we have $y=\textcolor{#00d865}{a}x^2$
• $\textcolor{#00d865}{a}$ can never be zero. In this case, we wouldn’t have an $x^2$ term and therefore the equation would not be a quadratic.
Level 4-5GCSEEdexcel iGCSE

To plot a quadratic graph let’s take a look at the following equation:

$y=x^2+7x+10$

You need to create an $\textcolor{#00d865}{xy}$ table and plot the coordinates

Substituting the values $x=-7$ to $x=0$, we get the following table:

Wondering how we got the values in the table?

When $x=-7 \rightarrow y=(-7)^2+7(-7)+10=10$ and so on for all values of $x$

Plotting these points as coordinates we get the graph seen to the right.

Note: Quadratic graphs won’t always be in the form of

$y=ax^2+bx+c$

In this scenario, you would need to rearrange the equation. For instance:

$5y-10x^2=15x+25 \rightarrow y=2x^2+3x+5$

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## Quadratic Graphs – Finding Solutions

To find a solution to a quadratic from a graph you take a look at where the graph crosses the $x$ axis. There are $3$ cases:

1. Doesn’t cross the $x$ axis at all
2. Touches the $x$ axis once
3. Crosses the $x$ axis at two points

Here are examples of the $3$ cases:

The solutions to each case are as follows:

1. No solutions
2. $x=2$
3. $x=-1$ and $x=-5$
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## Cubic Graphs – General Shape

Cubic graphs take the form

$y=\textcolor{#10a6f3}{a}x^3+\textcolor{#10a6f3}{b}x^2+\textcolor{#10a6f3}{c}x+\textcolor{#10a6f3}{d}$

Cubic graphs can either be positive ($a>0$) or negative ($a<0$)

Note:

• $\textcolor{#10a6f3}{b}, \textcolor{#10a6f3}{c}$ and $\textcolor{#10a6f3}{d}$ can be zero. In this case, we have $y=\textcolor{#10a6f3}{a}x^3$
• $\textcolor{#10a6f3}{a}$ can never be zero. In this case, we wouldn’t have an $x^3$ term and therefore the equation would not be a cubic.
Level 6-7GCSEEdexcel iGCSE

## Cubic Graphs – Plotting

To plot a cubic graph let’s take a look at the following equation:

$y=x^3-3x+1$

You need to create an $\textcolor{#10a6f3}{xy}$ table and plot the coordinates

Substituting the values $x=-2$ to $x=2$, we get the following table:

Wondering how we got the values in the table?

When $x=-2 \rightarrow y=(-2)^3-3(-2)+1=-1$ and so on for all values of $x$

Plotting these points as coordinates we get the graph seen to the right.

Note: Cubic graphs won’t always be in the form of

$y=ax^3+bx^2+cx+d$

In this scenario, you would need to rearrange the equation.

For instance:

$\dfrac{2x^4+3x^2}{y+2}=x \rightarrow \dfrac{2x^4+3x^2}{x}=y+2 \rightarrow y=2x^3+3x-2$

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## Cubic Graphs – Finding Solutions

To find a solution to a cubic from a graph you take a look at where the graph crosses the $x$ axis. There are $3$ cases:

1. Crosses the $x$ axis at one point
2. Crosses the $x$ axis at one point AND touches at a second point
3. Crosses the $x$ axis at three points

Here are examples of the $3$ cases:

The solutions to each case are as follows:

1. $x=2$
2. $x=-1$ and $x=1$
3. $x=-1$, $x=1$ and $x=3$
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## Reciprocal Graphs – General Shape (Type 1)

The first type of reciprocal graph takes the form

$y=\dfrac{\textcolor{#f95d27}{k}}{x}$

This type of reciprocal graph can either be positive ($k>0$) or negative ($k<0$)

Note:

• As $\textcolor{#f95d27}{k}$ becomes larger in both directions of positive and negative, the reciprocal graphs are plotted in the same shape but further away from the origin. For instance:
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## Reciprocal Graphs – General Shape (Type 2)

The second type of reciprocal graph takes the form

$y=\dfrac{\textcolor{#f95d27}{k}}{x^2}$

This type of reciprocal graph can either be positive ($k>0$) or negative ($k<0$)

Note:

• As $\textcolor{#f95d27}{k}$ becomes larger in both directions of positive and negative, the reciprocal graphs are plotted in the same shape but further away from the origin. For instance:
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## Reciprocal Graphs – Plotting (Type 1 and Type 2)

To plot a reciprocal graph let’s take a look at the following equation:

$y=\dfrac{4}{x}$

You need to create an $\textcolor{#f95d27}{xy}$ table and plot the coordinates

We can chose the values used in the $\textcolor{#f95d27}{xy}$ table, often it may be a wise choice to chose $x$ values that produce easy to plot $y$ values

Substituting the range of values in the table below:

Wondering how we got the values in the table?

When $x=-4 \rightarrow y=\dfrac{4}{-4}=-1$ and so on for all values of $x$

Plotting these points as coordinates we get the graph seen to the right.

Note: Reciprocal graphs never cross the $x$ axis therefore, don’t have any solutions. The same method applies to the type 2 reciprocal graphs.

Level 6-7GCSEEdexcel iGCSE

Plot the following equation on a set of $x$ and $y$ axes, then find the solutions to the quadratic.

$y=x^2-9x+14 \,\,\,\,\,\,\, 1\leq x \leq 8$

[4 marks]

Substituting the values $x=1$ to $x=8$, we get the following table:

Plotting these points as coordinates we get the following graph (as seen on the right).

From the plot we can see the graph crosses the $x$ axis at $2$ and $7$. These are the solutions to the quadratic.

Level 4-5GCSEEdexcel iGCSE

## Example 2: Cubic Graphs

Plot the following equation on a set of $x$ and $y$ axes, then find the solutions to the cubic.

$y=x^3-2x^2+x \,\,\,\,\,\,\, -1\leq x \leq 2$

[4 marks]

Substituting the values $x=-1$ to $x=2$, we get the following table:

Plotting these points as coordinates we get the following graph (as seen on the right).

From the plot we can see the graph crosses the $x$ axis at $0$ and touches at $1$. These are the solutions to the cubic.

Level 6-7GCSEEdexcel iGCSE

## Example 3: Reciprocal Graphs

Plot the following equation on a set of $x$ and $y$ axes,

$y=\dfrac{4}{x^2}\,\,\,\,\,\,\, -4\leq x \leq 4$

[4 marks]

Substituting the range of values in the table below:

Plotting these points as coordinates we get the following graph (as seen on the right).

Level 6-7GCSEEdexcel iGCSE

## Quadratic, Cubic and Harder Graphs Example Questions

Question 1: Plot the following equation on a set of $x$ and $y$ axes, then find the solutions to the quadratic.

$y=x^2+8x+15 \,\,\,\,\,\,\, -7\leq x \leq -1$

[4 marks]

Level 4-5GCSE Edexcel iGCSE

Substituting the values $x=-7$ to $x=-1$, we get the following table:

Plotting these points as coordinates we get the following graph:

From the plot we can see the graph crosses the $x$ axis at $-5$ and $-3$. These are the solutions to the quadratic.

Gold Standard Education

Question 2: Plot the following equation on a set of $x$ and $y$ axes, then find the solutions to the cubic.

$y=x^3-2x^2-2x-3 \,\,\,\,\,\,\, -1\leq x \leq 3$

[4 marks]

Level 6-7GCSE Edexcel iGCSE

Substituting the values $x=-1$ to $x=3$, we get the following table:

Plotting these points as coordinates we get the following graph:

From the plot we can see the graph crosses the $x$ axis at $3$. This is the solution to the cubic.

Gold Standard Education

Question 3: Plot the following equation on a set of $x$ and $y$ axes.

$y=\dfrac{5}{x}\,\,\,\,\,\,\, -4\leq x \leq 4$

[3 marks]

Level 6-7GCSE Edexcel iGCSE

Substituting the range of values in the table below:

Plotting these points as coordinates we get the following graph: