Graphing techniques
Learn one idea at a time
Choose a unit. Each has worked examples and practice.
Check the graph and its symmetries
Use the calculator as evidence alongside exact mathematical checks.
Recognise curves and asymptotes
Connect polynomial, exponential and rational forms to their graph features.
Transform and verify the whole curve
Reflect negative outputs and check the full curve, domain and turning points.
By the end, you can
- Sketch rational curves using structural features
- Find intercepts, turning points and asymptotes
- Explain why a calculator window can hide features
3 focused units · about 40 minutes for the full chapter. Allow extra time for practice.
The central idea
A sketch is a description of a function, not a screenshot. Find the domain, intercepts and symmetry, then explain its behaviour near excluded inputs and far from the origin. Polynomial division exposes horizontal or oblique asymptotes of rational functions.
Calculator entries for basic graphs
Enter each rule in Y=, use X,T,θ,n for the variable and GRAPH to draw it. The source calculator table gives:
| Function | Entry |
|---|---|
| variable, square | |
| variable, power, 3 | |
| 2nd, (square-root template), variable | |
| 1, divide, variable | |
| 1, divide, variable, square | |
| MATH, 4 (cube-root template), variable | |
| 2nd, LN (exponential template), variable | |
| exponential template, negative key, variable | |
| LN, variable |
The reciprocal entries also use ALPHA, Y= as a fraction-template alternative. To zoom into the cubic at , position the zoom centre there and use ZOOM, 2, ENTER. Another calculator model may label these operations differently. Check the expression, domain and window.
Worked example
For , the domain excludes 1. The vertical asymptote is , the horizontal asymptote is , and intercepts are and . The two branches approach the asymptotes with the sign of .
A graphing window does not prove that no other branch or turning point exists.
Check your understanding
A student sketches using only and claims that the curve has one turning point. Establish the original domain, both axial intercepts and the asymptotes. Use the derivative to assess the turning-point claim and explain what a more informative sketch must include.
The domain is . The x-intercepts are and ; the y-intercept is . Division gives , so the asymptotes are and .
Since , stationary inputs satisfy . The left branch has a local maximum at ; the right branch has a local minimum at , as the derivative sign changes confirm. The right input is approximately 3.73, outside the chosen window. A complete sketch must show both continuous branches, both turning points and the asymptotes, retaining the excluded input.
Practice questions
32 examples and practice questions, in 6 sections.
More practice
Before trusting a screen, predict the domain, branch behaviour and exact structural features. Explain any disagreement by checking both the rule and the window.
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