Recognise function graphs and read their features
Identify common graph families from their shape and connect parameters, symmetry, amplitude and period to visible features.
Section 3 · Functions
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which graph family has a constant gradient?
What you will learn
- Recognise linear and quadratic graphs for Core
- Recognise cubic, reciprocal, exponential and trigonometric graphs for Extended
- Read parameters, values and symmetry from a graph
Match the shape to a function family
A linear function y = ax + b has a constant gradient. A quadratic y = ax² + bx + c is a parabola; the sign of a controls whether it opens up or down.
Extended adds cubic S-shaped graphs, reciprocal branches, exponential growth or decay, and a sin(bx), cos(bx) or tan x graph. A reciprocal graph has two branches; an exponential graph stays on one side of its horizontal asymptote.
Read parameters and symmetry
Use intercepts, turning points and symmetry to identify coefficients. For y = a sin(bx) or y = a cos(bx), the amplitude is |a| and the period in degrees is 360°/|b|.
A graph may include zero coefficients, so it can simplify to a lower-degree form. Check a proposed rule against more than one point on the graph.
Pause and explain
What is the period of y = sin(2x), with x measured in degrees?
Worked example
For f(x) = x² − 4x + 3, find the roots, turning point and line of symmetry.
Show the worked solution
- Factor: f(x) = (x − 1)(x − 3), so roots are 1 and 3.
- Complete the square: f(x) = (x − 2)² − 1.
- The turning point is (2, −1), and the symmetry line is x = 2.
Answer Roots 1 and 3; minimum (2, −1); symmetry line x = 2.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Classify f(x) = −2x + 5 and g(x) = x² − 6x + 8. Find the roots of g and its symmetry line.
Give me a hint
Use the highest power to identify the family, then factor the quadratic.
Compare my reasoning
- f is linear; g is quadratic.
- g(x) = (x − 2)(x − 4), giving roots 2 and 4.
- The roots are symmetric about x = 3, so the axis is x = 3.
f linear; g quadratic with roots 2 and 4 and symmetry line x = 3.
Look for these in your work
- Families distinguished
- Roots found
- Symmetry line midway between roots
For y = 2 cos(3x) with x in degrees, state the amplitude and period. Name the basic shape of y = 1/x.
Give me a hint
Read the trig coefficient and use a full 360° cycle; recall the reciprocal graph's branches.
Compare my reasoning
- Amplitude = |2| = 2.
- Period = 360°/3 = 120°.
- y = 1/x is a reciprocal curve with branches in quadrants I and III.
Amplitude 2; period 120°; y = 1/x is reciprocal.
Look for these in your work
- Absolute coefficient used
- Period divided by frequency factor
- Reciprocal shape identified
A cooling model is plotted as temperature against time and falls quickly before flattening towards room temperature. Which graph family is a plausible model, and what feature should you check before adopting it?
Give me a hint
Compare the changing rate and long-run behaviour with the exponential family; use plotted data rather than shape alone.
Compare my reasoning
- An exponential decay curve is plausible because it changes quickly at first and then levels toward a horizontal value.
- Check whether the graph approaches the expected room-temperature level.
- Compare additional measured points; a visual resemblance alone does not establish a model.
Exponential decay is plausible; check its horizontal limiting level and agreement with data.
Look for these in your work
- Shape connected to a model
- Contextual limit checked
- Model not claimed from one visual cue
Common mistakes
- Calling a parabola linear because it has a straight-looking short section
- Reading a reciprocal branch as a line
- Using b as the amplitude in a trig graph
- Assuming a graph is symmetric without checking its axis
What can you explain now?
What is the amplitude of y = 3 sin(2x)?
Compare with the explanation
3
Amplitude is the absolute value of the coefficient of sine.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Which visible features distinguish linear, quadratic, reciprocal and exponential graphs, and how do a and b affect a trigonometric graph?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.