When asked to graph a function, the GDC does most of the heavy lifting — but you still need to recognise and label the key features: turning points, intercepts, asymptotes, symmetry. And the words “sketch” and “draw” mean different things — knowing which one is being asked for matters for marks.
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What you need to know
The graph of y = f(x) plots inputs on the x-axis and outputs on the y-axis
“Sketch” = freehand, key points labelled. “Draw” = ruler, to scale, accurate
Use your GDC to plot first, then transfer the key points to your sketch
Key features to label: turning points, x-intercepts, y-intercept, asymptotes, symmetry
Asymptotes are lines the graph approaches but never crosses — GDCs don’t always show them
“Sketch” vs “Draw” — They’re Not the Same
The wording matters. Reading a question carefully here is the difference between full marks and lost ones:
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Sketch
Freehand — doesn’t have to be exact
Axes labelled x and y
Correct general shape
Key points labelled with coordinates
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Draw
Pencil and ruler
Drawn to scale
Plot points accurately (table of values may help)
Join with smooth curve or straight lines
Quick rule: if a question says “sketch”, you don’t need a ruler — just label the key points. If it says “draw”, treat it like a precision exercise: ruler, scale, accurate points.
Using Your GDC to Sketch
Even when sketching freehand, the GDC is your starting point. The workflow is simple:
Plot the function on your GDC.
Find the key points using built-in tools (zero, max, min, intersect) — read off their coordinates.
Reproduce the shape from the GDC screen on paper, labelling all the key points.
Adjust the zoom! The default GDC window can hide turning points or intercepts. If something looks off, zoom out or change the window settings to make sure you’re seeing the whole shape.
The Key Features You Must Label
These are the things examiners want to see marked on your sketch:
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Turning Points
Where the graph changes direction (up ↔ down). Local max or min. For quadratics this is the vertex.
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Intercepts
x-intercepts (zeros/roots) where y = 0. y-intercept where x = 0.
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Symmetry
Some graphs reflect about a line. A quadratic, for example, has a vertical line of symmetry.
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Asymptotes
Lines the graph approaches but never touches. Can be horizontal or vertical. Easy to miss!
Anatomy of a Graph
Here’s a typical curve with all its key features labelled:
All the Features at a Glance
Asymptotes Up Close
Asymptotes are easy to miss because most GDCs don’t draw them. Always look at the equation:
Exponential
y = 2x
Horizontal asymptote at y = 0
Reciprocal
y = 1x
Both x = 0 and y = 0
Spot asymptotes from the equation: for y = 1x − 2, the denominator is zero at x = 2 — so there’s a vertical asymptote at x = 2. The horizontal one (y = 0) comes from the fact that 1/(huge number) → 0.
Worked Examples
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Example 1 — Identify key features of a quadratic
For f(x) = x2 − 4x − 5, use your GDC to find: (a) the vertex, (b) the x-intercepts, (c) the y-intercept.
Answer:
(a) Plot on GDC, use the minimum function.Vertex: (2, −9)(b) Use the zero/root function.x² − 4x − 5 = (x − 5)(x + 1) = 0x-intercepts: (−1, 0) and (5, 0)(c) Substitute x = 0.f(0) = 0 − 0 − 5 = −5y-intercept: (0, −5)
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Example 2 — Draw the graph (accurate)
Draw the graph of y = x2 − 4x − 5 for −2 ≤ x ≤ 6.
Answer:
“Draw” means accurately. Use GDC to find vertex, roots, y-intercept, then plot to scale.Vertex: (2, −9)Roots: (−1, 0), (5, 0)y-intercept: (0, −5)Drawn to scale with all key points labelled
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Example 3 — Sketch a graph with asymptotes
Sketch the graph of g(x) = 2 + 1x + 1.
Answer:
“Sketch” means rough — but show key points. Use GDC for intercepts and check the equation for asymptotes.x-intercept: set g(x) = 0.2 + 1/(x+1) = 0 → 1/(x+1) = −2x + 1 = −1/2 → x = −3/2x-intercept: (−3/2, 0)y-intercept: substitute x = 0.g(0) = 2 + 1/1 = 3y-intercept: (0, 3)Asymptotes from equation:denominator x + 1 = 0 → vertical asymptote x = −1as x → ∞, the fraction → 0 → horizontal asymptote y = 2Asymptotes: x = −1 and y = 2Sketch the curve approaching but never touching the dashed asymptote lines.
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Example 4 — Sum of two functions
Given f(x) = x2 and g(x) = 2x − 1, find the local minimum of y = f(x) + g(x) on your GDC.
Answer:
Step 1: write the combined function.y = x² + 2x − 1Step 2: enter into GDC and use the minimum function.Minimum at (−1, −2)For a quadratic, this is the vertex. The GDC handles sums and differences directly.
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Example 5 — A graph with multiple turning points
For f(x) = x3 − 3x, use your GDC to find the local maximum, the local minimum, and all x-intercepts.
Answer:
Plot on GDC. Use max, min, and zero functions.x-intercepts: factorise.x³ − 3x = x(x² − 3) = 0x = 0, x = ±√3x-intercepts: (−√3, 0), (0, 0), (√3, 0)Local max (left peak):(−1, 2)Local min (right valley):(1, −2)A cubic can have up to 2 turning points and up to 3 x-intercepts. Always check both.
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Tips
Read the verb carefully. “Sketch” lets you go freehand; “draw” demands a ruler and accuracy. Don’t waste time using a ruler when freehand is fine.
Always use the GDC first. Plot the function, find the key points using the built-in tools, then transfer onto paper.
Sketch the GDC graph as part of your working even when not explicitly asked. It’s good exam technique and earns method marks.
Check the equation for asymptotes — most GDCs don’t draw them. Look for division by zero (vertical) or the long-run behaviour (horizontal).
Adjust the GDC window when the default view hides features. Zoom out, then zoom in around interesting points.
Label everything — coordinates of all turning points, intercepts, and asymptote equations.
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Common mistakes
Missing the asymptotes. Most GDCs don’t show them. If the function has a fraction or an exponential, check the equation manually.
Confusing local and global. A graph can have several local maxes/mins. The biggest/smallest of all is the global one.
Forgetting to label coordinates. Saying “the curve has a maximum here” without writing (x, y) loses marks.
Treating sketch like draw. Wasting time using a ruler and grid for a sketch question — clean curves with labelled points are enough.
Not labelling the axes. Even on a sketch, both axes need x and y written next to the arrows.
Drawing the curve through an asymptote. The graph approaches but never crosses a vertical asymptote — your sketch must show that.
Final word: Plot on GDC, find the key points, transfer onto paper, label everything. Watch the verb (sketch vs draw), and always check the equation for asymptotes the GDC won’t show you.
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