Curve Sketching

What is Curve Sketching?

Curve Sketching is an essential skill within mathematics where we draw a curve or graph from a given equation. This technique is not only integral to understanding mathematical concepts and solving equations, but it’s also pivotal in real-world applications. Engineers, architects, data scientists, statisticians, and economists routinely utilize curve sketching in their work. For example, it assists engineers in modelling stress-strain curves in materials, helps architects design curved structures, plays a crucial role in data visualisation in data analytics and allows economists to plot supply-demand curves. Ultimately, mastering curve sketching allows us to visually represent complex equations and utilise this understanding for practical problem-solving across various industries.

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.

See Example 1.

Approach 2 – Applying Transformations to a known curve

Alternatively 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 Cubics) and some quartics (see above). Note that the shapes of the basic polynomials are as follows:

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}:
curve sketching

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

curve sketching

See Example 2.

Examples

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

Solution:

1. Firstly, the general shape is seen here on the right.

2. Next, find the y-intercept by substituting x=0 – the curve crosses at the origin.

3. Then, find the roots by substituting y=0 and solving for x. It follows that x=0 is a repeated root, as is x=0.5.

4. Since we know where the roots are, we can place the graph in the correct location.

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

Solution:

1. Firstly, we start with the graph of y=\frac{1}{x} and apply appropriate transformations. The graph y=\frac{1}{x} has asymptotes at x=0 and y=0.


2. Next we can identify a multiplication of 2: f(x)\rightarrow 2f(x) which is a stretch in the y-direction. Call this new graph g(x) and it looks very much like the original. The asymptotes are as before.


3. Then replace x with x-5 i.e. g(x)\rightarrow g(x-5), this is a shift to the right in the x-direction. Call this h(x). The asymptotes are now at x=5 and y=0.

4. Finally subtract 3 from this graph which is a shift downwards in the y-direction: h(x)\rightarrow h(x)-3

5. The following graph is the final graph showing y=\frac{2}{x-5}-3. The asymptotes are at x=5 and y=-3.

Videos

https://youtu.be/wDquyccgrHA

Sketching a cubic then performing a transformation from an unknown constant.

https://youtu.be/eX0s60rR87k

Explanation for why the graph of y=\frac{1}{x^2} looks the way it does including an introduction to odd and even functions.

Extra Resources

Sketching Cubics Exercises

Cubics

Open Cubic Sketching Exercises in Neww Window

Transformations Exercises

Transformations

Open Transformation Exercises in New Window

Questions by Topic

AS Maths Algebra & Functions

A2 Maths Algebra & Functions