### Sketching Graphs

#### Sketching

Graphs have a horizontal $x$-axis and vertical $y$-axis, and the point where the two axes cross is the origin. A point on a graph has coordinate $(x,y)$. The easiest way to sketch a function $f(x)$ is to choose some values of $x$ and put them into the function to find the corresponding $y$ values. This will give a set of coordinates to be plotted which can then be joined with a line or curve. Most of the time, only a rough sketch will be necessary. This means that the graph just needs to be of the right shape with important points, such as where it cuts the axes, labelled.

#### Useful Graphs

It is important to know some graphs straight away without having to work out points to plot first. The most common are:

#### Transformations

Transformations map all the points of a function onto a new function. Transformations include translating, scaling and reflecting.

##### Translations

A translation is the movement of the graph a certain number of units left, right, up or down.

Example: take the graph $f(x) = x^2$.

A translation $a$ units in the positive $x$ direction is defined by the function $f(x-a) = (x-a)^2$.

A translation $a$ units in the negative $x$ direction is defined by the function $f(x+a) = (x+a)^2$.

A translation $a$ units in the positive $y$ direction is defined by the function $f(x) +a= x^2 + a$.

A translation $a$ units in the negative $y$ direction is defined by the function $f(x) -a= x^2 - a$.

##### Scaling

Scaling is the process of stretching or squashing a graph in a certain direction.

Example: take the graph $f(x) = x^2$.

Scaling by a factor of $a$ in the $x$ direction is defined by the function $f\left(\dfrac{1}{a}x\right) = \left(\dfrac{1}{a}x\right)^2$.

Scaling by a factor of $\dfrac{1}{a}$ in the $x$ direction is defined by the function $f(ax) = (ax)^2$.

Scaling by a factor of $a$ in the $y$ direction is defined by the function $af(x) = ax^2$.

Scaling by a factor of $\dfrac{1}{a}$ in the $y$ direction is defined by the function $\dfrac{1}{a}f(x) = \dfrac{1}{a}x^2$.

##### Reflections

Reflecting involves taking all the points in the graph and reflecting them over the $x$ or $y$ axis.

Example: take the graph $f(x) = x^2$.

Reflecting in the $x$ axis is defined by the function $-f(x) = -(x^2)$.

Reflecting in the $y$ axis is defined by the function $f(-x) = (-x)^2$.

#### Video Examples

###### Example 1

Prof. Robin Johnson sketches the graph of $y=3x^2+6x$.

###### Example 2

Prof. Robin Johnson sketches the graph of $y=x+\dfrac{1}{x}$.

###### Example 3

Prof. Robin Johnson sketches the function f(x) = \begin{cases}\begin{align} &1+x, \;\; x \geq 0\\&1-2x, \, x\lt0 \end{align}\end{cases}.

#### Workbook

This workbook produced by HELM is a good revision aid, containing key points for revision and many worked examples.