## Approach 1 – Steps for Curve Sketching

Follow these steps for sketching a curve:

1. Firstly, identify the general shape of the curve and whether it is of a negative or positive shape.
2. Next, find the y-intercept – substitute x=0 into the equation of the graph to see where the graph cuts the y-axis.
3. Then, identify the roots of the cubic – this is where the graph should cut the x-axis. This may involve factorising and you should note that the graph will bounce off the x-axis at any repeated roots.
4. Finally, place the curve so that it cuts the x and y axes at the correct points making sure that the curve touches the x axis at any repeated roots.

EXAMPLE

Sketch the quartic $y=x^2(2x-1)^2$.

See Cubics to find out more about sketching cubics.

## Approach 2 – Applying Transformations to a known curve

Alternative to the above approach you may be asked to sketch a curve by performing transformations to a curve you already know or one that is given to you. Click here to see the various Transformations that you should know how to perform.

### Polynomials

You should know how to sketch some polynomials including Quadratics (see Quadratics or Completing the Square), Cubics (click here to see Cubic Sketching) and some Quartics (see above). Note that that shapes of the basic polynomials are as follows but they can each have central stationary points accordingly:See Completing the Square to see how to use Transformations to sketch a quadratic.

### Trigonometric Functions

Click here to see Trigonometric Graphs and some of their transformations.

### Reciprocals

Reciprocals are curves that have asymptotes (lines that are approached but never touched) due to division by x, see example below. Note that curves with equation $y=\frac{1}{x-a}+b$ have horizontal and vertical asymptotes. These asymptotes have equations $x=-a$ (vertical) and $y=b$ (horizontal). You may be required to know the following graphs and to perform transformations to them. Both curves have asymptotes at $x=0$ and $y=0$.

$\hspace{25pt}y=\frac{1}{x}$:

$\hspace{25pt}y=\frac{1}{x^2}$:

EXAMPLE

Sketch the graph of $y=\frac{2}{x-5}-3$.

## Curve Sketching Exercises

Cubics

Cubics

Cubics Questions – open in new window – solutions on the second page.

Click here to see some examples of curve sketching exercises for Oxbridge interviews. It is important to note that in Oxbridge interviews, you don’t necessarily have to get the correct answer – they are looking to see how you approach the question and your problem solving skills.

Alternatively, click here to find Questions by Topic and scroll down to all past TRANSFORMATIONS questions to practice some more curve sketching.

Are you ready to test your Pure Maths knowledge? If so, visit our Practice Papers page and take StudyWell’s own Pure Maths tests.