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**Learning Goals**

- Identify and understand harder graphs

Maths

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Graphs such as cubic graphs and reciprocal graphs are deemed more complicated than quadratic and linear graphs. In truth, they just have different shapes and hence other things to look out for.

The general cubic equation is

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

where $a$, $b$, $c$ and $d$ are constants. Their graphs have a kink in the middle. Below are some examples:

Above is a *positive* cubic. It starts low on the left and eventually moves up as it goes to the right. A positive cubic has a positive $a$ in the equation.

Above is a *negative* cubic. It starts high on the left and eventually moves down as it goes to the right. A negative cubic has a negative $a$ in the equation.

The kink in the middle does not have to be so pronounced - it may be flat:

One method to draw them is to use a table of coordinates. This is done in the same way as with linear graphs and quadratic graphs: by inserting given coordinates into the equation to find the corresponding coordinates.

*By completing the table of coordinates and plotting the points, sketch the cubic graph $y=x^3$.*

$x$ | $-3$ | $-2$ | $-1$ | $0$ | $1$ | $2$ | $3$ |

$y$ | | | | | | | |

*The cubic equation says to cube the $x$-coordinate.*

*Note: **when cubing a negative number, you get a negative number back.*

*Starting with $x=-3$, you have that *

*$y=x^3=(-3)^3=-27$*

*Continuing in this way, you get the following table of coordinates:*

$x$ | $-3$ | $-2$ | $-1$ | $0$ | $1$ | $2$ | $3$ |

$y$ | $-27$ | $-8$ | $-1$ | $0$ | $1$ | $8$ | $27$ |

*In the equation of the curve, the *$a$*-value is *$1$*, which is positive, so this curve will look more like the first image above. Plotting the points and fitting a suitable curve to them gives:*

*The dotted lines are shown for illustrative purposes. *

The reciprocal of a number is $1$ divided by that number. For example, the reciprocal of $3$ is $\frac13$. Reciprocal graphs have equations of the form

$y=\frac{A}{x}$

where $A$ is a constant. This could be rearranged to $x=\frac{A}{y}$ and in either form, it is clear that neither $x$ nor $y$ may equal zero, since you cannot divide $A$ by zero. Hence, the graph won't cross either axes. The graph of $y=\frac2x$ is given below:

If $A$ is positive in the $y=\frac{A}x$ equation, the graph will look very similar to this above. It has curves in quadrant one and quadrant three that get closer and closer to the axes without touching them.

The graph of $y=-\frac2x$ is given below:

If $A$ is negative in $y=\frac{A}{x}$, then the graph will look very similar to *this *above. It has curves in quadrant two and four that never touch the axes, but get forever closer.

Drawing a reciprocal graph can be achieved following the same table method as seen above with cubics.

Graphs such as cubic graphs and reciprocal graphs are deemed more complicated than quadratic and linear graphs. In truth, they just have different shapes and hence other things to look out for.

The general cubic equation is

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

where $a$, $b$, $c$ and $d$ are constants. Their graphs have a kink in the middle. Below are some examples:

Above is a *positive* cubic. It starts low on the left and eventually moves up as it goes to the right. A positive cubic has a positive $a$ in the equation.

Above is a *negative* cubic. It starts high on the left and eventually moves down as it goes to the right. A negative cubic has a negative $a$ in the equation.

The kink in the middle does not have to be so pronounced - it may be flat:

One method to draw them is to use a table of coordinates. This is done in the same way as with linear graphs and quadratic graphs: by inserting given coordinates into the equation to find the corresponding coordinates.

*By completing the table of coordinates and plotting the points, sketch the cubic graph $y=x^3$.*

$x$ | $-3$ | $-2$ | $-1$ | $0$ | $1$ | $2$ | $3$ |

$y$ | | | | | | | |

*The cubic equation says to cube the $x$-coordinate.*

*Note: **when cubing a negative number, you get a negative number back.*

*Starting with $x=-3$, you have that *

*$y=x^3=(-3)^3=-27$*

*Continuing in this way, you get the following table of coordinates:*

$x$ | $-3$ | $-2$ | $-1$ | $0$ | $1$ | $2$ | $3$ |

$y$ | $-27$ | $-8$ | $-1$ | $0$ | $1$ | $8$ | $27$ |

*In the equation of the curve, the *$a$*-value is *$1$*, which is positive, so this curve will look more like the first image above. Plotting the points and fitting a suitable curve to them gives:*

*The dotted lines are shown for illustrative purposes. *

The reciprocal of a number is $1$ divided by that number. For example, the reciprocal of $3$ is