๐ Purpose: Learn what a graph shows, how to sketch a function from algebraic info or a screen, what features to label, and how to use technology to graph functions and their sums/differences. Emphasis on clarity: every sketch must label axes and key features and justify choices.
| Term | Definition / Note |
|---|---|
| Graph of a function | Set of points (x, f(x)) in the plane. Visual summary of domain, range, continuity, intercepts, turning points and end behaviour. |
| Sketch vs draw | โSketchโ = approximate shape with key features labelled (no exact plotting needed). โDrawโ = careful plotting with scale and accurate coordinates. |
| Key features | x- & y-intercepts, local maxima/minima (turning points), asymptotes (vertical/horizontal/oblique), discontinuities, intervals of increase/decrease, symmetries (even/odd), periodicity. |
| Sum / difference of functions | (f + g)(x) = f(x) + g(x). Graphically add vertical coordinates pointwise (use technology or table of values for accuracy). |
๐ Sketching principles โ what to include and why
โข Label both axes and include scale marks where relevant.
โข Always mark and label: intercepts, turning points (approximate coordinates), vertical & horizontal asymptotes (equations), and any discontinuities (open/solid circles).
โข State whether the sketch is exact or approximate (sketch). If transferring from screen to paper, note the scale you used.
โข Use arrows on ends to indicate end behaviour when graph continues beyond drawn region (e.g., y โ โ as x โ โ).
๐ Creating a sketch from algebraic or contextual information
- Identify domain/range & continuity: look for roots, denominators, square roots, logs and their restrictions.
- Find intercepts: set x = 0 for y-intercept, solve f(x) = 0 for x-intercepts (approximate if needed).
- Find turning points (if easy): for simple polynomials use calculus (fโฒ(x)=0) or symmetry for quadratics. For SL, approximate coordinates are sufficient if labelled.
- Locate asymptotes: vertical where denominator 0 (and limit โ ยฑโ), horizontal/oblique from end behaviour (limits as x โ ยฑโ).
- Combine the info into shape: sketch curve, label features and indicate approximate coordinates or equations (e.g., x = 2, y = โ1).
๐ Examples & real situations
Example 1 โ Projectile height (context):
A ball thrown upwards: h(t) = โ4.9t2 + 20t + 1 (height in m, t in s).
Key features: domain t โฅ 0 until h(t) = 0 (impact time). Vertex at t = โb/(2a) = โ20/(2ร(โ4.9)) โ 2.04 s (maximum height โ h(2.04)). Sketch: label t-axis (time), h-axis (height), vertex coordinate โ (2.04, h(2.04)), and intercepts.
Meaning: Vertex = time of maximum height; zeros = impact times; end behaviour irrelevant (physical domain restricted to t โฅ 0).
Example 2 โ Hyperbola & asymptotes (situation):
f(x) = 1/(x โ 2). Vertical asymptote at x = 2 (denominator 0). Horizontal asymptote y = 0 (as x โ ยฑโ). Sketch both branches, label asymptotes and show behaviour near x = 2 (โ ยฑโ). If transferring from screen, read points like (3,1), (2.5,2) to guide shape.
Meaning: Asymptote lines are guides the graph approaches; discontinuity at x=2 (not in domain).
Example 3 โ Sums/differences of graphs:
f(x) = sin x, g(x) = 0.5x. To sketch (f + g)(x), make a table of x-values (e.g., x = โฯ, โฯ/2, 0, ฯ/2, ฯ), compute f(x) and g(x), add to get (f+g)(x) and plot. Technology: plot both f and g, then plot f+g to see how sinusoid is tilted by linear term.
Meaning: The linear term shifts the sinusoid vertically as x increases โ useful in modelling periodic processes with trend (e.g., seasonal sales with increasing baseline).
๐ Sketching sums & differences โ practical method
- Use table-of-values for f and g at same x-values, compute f ยฑ g pointwise, then plot or use GDC to plot f+g directly.
- When sketching by hand, choose 6โ8 representative x-values (including turning points and intercepts) and connect smoothly.
- Label points used (e.g., x = 0, ยฑ1, turning points) so examiner sees your method.
๐งฎ GDC Tips
- Use Table or Trace to read exact coordinates for key points when transferring to paper.
- Turn on grid & set axis limits so the region of interest is centered and clear.
- Plot f, g, and fยฑg in different colors; hide unnecessary curves when sketching by hand to avoid confusion.
๐ง Examiner Tip
Label axes & scale, indicate approximate coordinates for turning points/intercepts, show asymptote equations, and state whether the sketch is exact or approximate.
When using technology, include a short note describing how you extracted coordinates (table/trace) to justify approximations.
๐ TOK Perspective
Is a sketch (an idealized representation) as rigorous as an algebraic derivation?
Consider how visualization can both reveal patterns quickly and hide subtle algebraic constraints (e.g., domain restrictions).
Discuss the role of representation and the trade-off between speed and rigour in mathematical communication.
๐ Paper tips
- In exams: draw axes, label units if given, mark key points and write short justifications (e.g., “vertical asymptote x = 2 because denominator = 0”).
- When sketching sums/differences show the table of values or state you used software and list 3โ4 sampled points used to create the sketch.
โ ๏ธ Common pitfalls
- Not labelling axes or forgetting units โ makes interpretation ambiguous.
- Confusing sketch with exact graph; examiners expect approximate coordinates but clear labelling.
- Forgetting to show asymptote equations or open/closed circles at discontinuities.
๐ Practice questions (sketch & interpret)
- Sketch: y = x2 โ 4x + 3. Show vertex, intercepts and axis of symmetry. State domain & range.
- Sketch & transfer: Plot f(x) = tan x in region x โ (โฯ/2, ฯ/2). Label asymptotes and key points; transfer the central branch to paper using table-of-values.
- Sums: Given f(x) = eโx and g(x) = 0.5, sketch f and g and then sketch f + g. Explain how addition changes end behaviour.
- Context: A city’s traffic flow vs time is modelled as a periodic curve plus a linear trend. Sketch a plausible shape and explain turning points in context.
๐ Quick sketch checklist
- Axes labelled, scale marked, and units (if any) shown.
- Intercepts and approximate coordinates of turning points labelled.
- Asymptotes drawn with equations (dashed lines) and discontinuities shown as open circles.
- State whether sketch is exact or approximate; include 2โ4 reference points used to draw the curve.