# Quadratic, Cubic and Harder Graphs

## Quadratic, Cubic and Harder Graphs Revision

**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**

Quadratic graphs take the form

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.

**Quadratic Graphs – Plotting**

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

**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:

- Doesn’t cross the x axis at all
- Touches the x axis once
- Crosses the x axis at two points

Here are examples of the 3 cases:

The solutions to each case are as follows:

- No solutions
- x=2
- x=-1 and x=-5

**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.

**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

**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:

- Crosses the x axis at one point
- Crosses the x axis at one point
**AND**touches at a second point - Crosses the x axis at three points

Here are examples of the 3 cases:

The solutions to each case are as follows:

- x=2
- x=-1 and x=1
- x=-1, x=1 and x=3

**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:

**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:

**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.

**Example 1: Quadratic Graphs**

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.

**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.

**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).

## 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]**

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.

**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]**

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.

**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]**

Substituting the range of values in the table below:

Plotting these points as coordinates we get the following graph: