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International Mathematics 06072.1

Sketching and solving graphically

Using a graphics display calculator to sketch, solve and find key features.

Learning objectives

What you need to be able to do

Teacher-mapped phrasing — check against the official Cambridge syllabus for exact wording.

  • 2.1.1Sketch and interpret graphs of functions, and use a graphics calculator to find intersections, roots and turning points.

6 minute read

Sketching and solving graphically

0607 assumes a graphics display calculator (GDC), and questions are written expecting you to use it well. That changes technique rather than removing the need to understand.

What the GDC is for

  • Finding where a curve crosses the x-axis (roots/zeros).
  • Finding intersections of two graphs — which is how you solve equations graphically.
  • Finding maximum and minimum points.
  • Producing a sketch you can then describe.

Solving equations graphically

To solve f(x) = g(x), plot both and find the intersection — this is often faster and more reliable than algebra, and for many 0607 equations it is the intended method. Alternatively rearrange to f(x) − g(x) = 0 and find the zeros of that single function.

Sketching well

A sketch is not a plot. It must show:

  • Axis intercepts — where the curve crosses each axis;
  • Turning points;
  • Asymptotes, drawn as dashed lines;
  • The correct overall shape and behaviour at the extremes.

Label the axes and mark key coordinates. Marks come from these features, not from artistic accuracy.

Choosing a sensible window

The commonest GDC error is a window that hides the interesting part of the graph. If a graph looks blank or like a straight line, zoom out first, then narrow in. Always sanity-check that what you see matches what the equation should do.

Asymptotes

For a function such as y = 1/(x − 3), there is a vertical asymptote at x = 3 (division by zero) and a horizontal asymptote at y = 0. The GDC may draw a misleading near-vertical connecting line at the asymptote — that line is an artefact, not part of the graph.

Think of it like this

The GDC is a microscope: it shows you exactly what you point it at, so a badly chosen window is like examining the wrong slide. The instrument is only as good as the judgement directing it.

Worked examples

Method, step by step

Explain how you would use a graphics calculator to solve x³ − 2x = 1.

  1. 1Rearrange so one side is zero: x³ − 2x − 1 = 0.
  2. 2Plot y = x³ − 2x − 1 on the GDC.
  3. 3Choose a window wide enough to show all turning points and axis crossings.
  4. 4Use the zero/root function to find each x-intercept — these are the solutions.

Plot y = x³ − 2x − 1 and use the calculator's zero function to find each x-intercept; each root is a solution of the original equation.

Common misconceptions

  • Trusting the default window. A graph that looks blank usually means the window is wrong, not that the function is undefined.
  • Treating the vertical line a GDC draws at an asymptote as part of the curve. It is a plotting artefact.
  • Giving a sketch with no labelled intercepts or turning points — those features are where the marks are.

In the exam

  • When a question says "solve graphically" or gives an awkward-looking equation, plot both sides and find the intersection rather than attempting algebra.
  • Write down coordinates from the GDC to the accuracy the question requests, usually 3 significant figures, and state what each point represents.