Approach 1 – Steps for Curve Sketching
Follow these steps for sketching a curve:
- Firstly, identify the general shape of the curve and whether it is of a negative or positive shape.
- Next, find the y-intercept – substitute into the equation of the graph to see where the graph cuts the -axis.
- Then, identify the roots of the cubic – this is where the graph should cut the -axis. This may involve factorising and you should note that the graph will bounce off the -axis at any repeated roots.
- Finally, place the curve so that it cuts the and axes at the correct points making sure that the curve touches the 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.
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:
Click here to see Trigonometric Graphs and some of their transformations.
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 have horizontal and vertical asymptotes. These asymptotes have equations (vertical) and (horizontal). You may be required to know the following graphs and to perform transformations to them. Both curves have asymptotes at and .
See Example 2.
Sketching a cubic then performing a transformation from an unknown constant.
Explanation for why the graph of looks the way it does including an introduction to odd and even functions.