IGCSE Maths · Algebra and Graphs · Core & Extended

Graphs of Functions: Tables of Values, Curves and Solving Graphically

Every function in this note is built the same way: substitute a range of x-values in, get a table of y-values out, plot the points, and join them into a curve. Core covers linear, quadratic, and reciprocal graphs; Extended covers a much wider family, including cubics, other power graphs, and exponentials — tagged throughout.

1. Constructing Tables of Values

Choose a sensible range of x-values (usually given in the question), substitute each into the function, and record the results — exactly the substitution skill from the Introduction to Algebra note, just repeated systematically.

Worked example Construct a table of values for y = x² − 2x − 3, for x from −2 to 4.
x−2−101234
y50−3−4−305
Each y-value comes from substituting: e.g. at x = −2, y = (−2)² − 2(−2) − 3 = 4 + 4 − 3 = 5.
Try it — Tables of Values
  1. Construct a table of values for y = 2x + 1, for x from −2 to 2.
  2. Construct a table of values for y = x² + 1, for x from −2 to 2.
Show answers
  1. x: −2,−1,0,1,2 → y: −3,−1,1,3,5
  2. x: −2,−1,0,1,2 → y: 5,2,1,2,5

2. Quadratic Graphs (y = ±x² + ax + b)

A quadratic graph is a smooth U-shaped curve (or an upside-down ∩-shape if the x² coefficient is negative), called a parabola. It’s always symmetrical about a vertical line through its turning point.

Plotting y = x² − 2x − 3, using the table from Section 1 x y −1 3 (1, −4) The curve crosses the x-axis at x = −1 and x = 3 — these are the roots of x² − 2x − 3 = 0. The lowest point (turning point) is at (1, −4), and the curve is symmetrical about the vertical line x = 1.
Try it — Quadratic Graphs
  1. For y = x² − 4x + 3, find the y-values when x = 0, 1, 2, 3, 4.
  2. Does y = −x² + 2x + 3 open upward (U-shape) or downward (∩-shape)? Explain how you know.
  3. Using your table from question 1, at which two x-values does the graph cross the x-axis?
Show answers
  1. x=0: 3. x=1: 0. x=2: −1. x=3: 0. x=4: 3
  2. Downward (∩-shape) — the coefficient of x² is negative.
  3. x = 1 and x = 3 (where y = 0 in the table)

3. Reciprocal Graphs (y = a/x)

A reciprocal graph has two separate curved branches that never touch the axes — the axes are called asymptotes, lines the curve gets closer and closer to but never actually reaches.

Plotting y = 3/x x y (1,3) (−1,−3) Notice the two branches sit in opposite corners (top-right and bottom-left, since a = 3 is positive) and curve away from both axes without ever crossing them. If a were negative instead, the branches would sit in the other two corners.
Try it — Reciprocal Graphs
  1. For y = 4/x, find y when x = 1, 2, 4, −1, −2.
  2. For y = −2/x, in which two corners (quadrants) would the branches of the graph sit?
  3. Explain why the graph of y = a/x can never cross the y-axis.
Show answers
  1. x=1: 4. x=2: 2. x=4: 1. x=−1: −4. x=−2: −2
  2. Top-left and bottom-right (since a is negative, when x is positive y is negative, and vice versa).
  3. The y-axis is where x = 0, and a/0 is undefined — there’s simply no valid y-value there.

4. Solving Equations Graphically

Where a graph crosses the x-axis gives the roots of the equation (solutions where y = 0). Where two graphs cross each other gives the solution to the equation formed by setting them equal.

Worked example — finding roots Use the graph of y = x² − 2x − 3 to solve x² − 2x − 3 = 0.
Read off where the curve crosses the x-axis (shown in the Section 2 diagram): x = −1 or x = 3
Worked example — finding an intersection (−1, 0) (4, 5) Use the graph to solve x² − 2x − 3 = x + 1, by finding where the curve y = x² − 2x − 3 meets the line y = x + 1.
Reading off the two intersection points: x = −1 or x = 4
Check algebraically: x² − 2x − 3 = x + 1 → x² − 3x − 4 = 0 → (x−4)(x+1) = 0 → x = 4 or x = −1 ✓
Try it — Solving Graphically
  1. A graph of y = x² − 4x + 3 crosses the x-axis at two points. Given the table from Section 2’s practice, what are they?
  2. Explain, in words, how you would use a graph to solve x² + x − 6 = 2x, if you already had the graph of y = x² + x − 6 drawn.
Show answers
  1. x = 1 and x = 3
  2. Draw the line y = 2x on the same axes, and read off the x-coordinates of the points where the line crosses the curve.

5. Cubic Graphs Extended only

A cubic graph (y = ax³ + … ) has a distinctive S-shaped curve, typically with two turning points — one local maximum and one local minimum.

Plotting y = x³ − 3x local max (−1, 2) local min (1, −2) The curve rises, turns at a local maximum, falls through the origin, turns again at a local minimum, then rises again — this “wiggle” shape (two turning points) is characteristic of cubic graphs where the x³ coefficient is positive.
Try it — Cubic Graphs Extended
  1. For y = x³, find y when x = −2, −1, 0, 1, 2.
  2. How many times does a typical cubic graph (with two turning points) cross the x-axis, at most?
Show answers
  1. x=−2: −8. x=−1: −1. x=0: 0. x=1: 1. x=2: 8
  2. Three times (three real roots).

6. Other Power Graphs Extended only

The Extended syllabus allows functions built from axⁿ for a range of powers n, including negative and fractional ones — the method for plotting them is identical: build a table of values, then plot.

FunctionShape
y = a/x² (n = −2)Two branches, both in the upper half if a is positive (since squaring removes negative signs)
y = a√x (n = 1/2)Starts at the origin, curves upward, only defined for x ⩾ 0
Worked example Construct a table of values for y = 2√x, for x = 0, 1, 4, 9.
x=0: y=0. x=1: y=2. x=4: y=4. x=9: y=6.
Plotting these shows a curve that starts at the origin and rises, getting less steep as x increases.
Try it — Other Power Graphs Extended
  1. For y = 4/x², find y when x = 1, 2, 4.
  2. For y = 3√x, find y when x = 0, 1, 4, 9.
Show answers
  1. x=1: 4. x=2: 1. x=4: 0.25
  2. x=0: 0. x=1: 3. x=4: 6. x=9: 9

7. Exponential Graphs Extended only

An exponential function has the form y = ab^x + c, where the variable is in the exponent. These graphs model growth or decay — the curve never crosses its horizontal asymptote.

Plotting y = 2ˣ (0,1) (2,4) approaches y = 0, never touches As x increases, y grows faster and faster (this is exponential growth). As x decreases into negative values, the curve flattens and approaches y = 0 without ever quite reaching it — the x-axis is a horizontal asymptote here.
Worked example — exponential decay A population of bacteria is modelled by P = 200 × (0.5)ᵗ, where t is time in hours. Construct a table of values for t = 0, 1, 2, 3, and describe the shape of the graph.
t=0: P=200. t=1: P=100. t=2: P=50. t=3: P=25.
The graph falls steeply at first, then flattens — this is exponential decay, and the curve approaches P = 0 but never reaches it.
Try it — Exponential Graphs Extended
  1. For y = 3ˣ, find y when x = 0, 1, 2, 3.
  2. For y = 100 × (0.8)ˣ, find y when x = 0, 1, 2 (round to 1 d.p. where needed).
  3. Does y = 5ˣ represent growth or decay? What about y = (0.2)ˣ?
Show answers
  1. x=0: 1. x=1: 3. x=2: 9. x=3: 27
  2. x=0: 100. x=1: 80. x=2: 64
  3. y = 5ˣ is growth (base > 1). y = (0.2)ˣ is decay (base between 0 and 1).
Common mistakes to watch for
  • Plotting points but joining them with straight line segments for a curved function — quadratics, cubics, reciprocals and exponentials should all be smooth curves, not straight-edged shapes.
  • Forgetting a reciprocal graph is undefined at x = 0 — never plot or join a point there.
  • Confusing “roots” (where a curve crosses the x-axis) with the y-intercept (where it crosses the y-axis) — they answer different questions.
  • Assuming every quadratic crosses the x-axis twice — some only touch it once, and some (if they don’t reach y = 0 at all) don’t cross it at all.
  • Mixing up growth and decay for exponentials — check whether the base is greater than 1 (growth) or between 0 and 1 (decay).

8. Quick-Fire Challenge Round

True or False?
  1. A quadratic graph is always symmetrical.
  2. A reciprocal graph y = a/x touches the origin.
  3. The roots of a quadratic are where the graph crosses the x-axis.
  4. An exponential graph y = abˣ (with b > 1) eventually crosses the x-axis for large enough x.
Show answers
  1. True — every parabola is symmetrical about a vertical line through its turning point.
  2. False — it never touches either axis; both are asymptotes.
  3. True.
  4. False — it gets closer and closer to the x-axis as x decreases, but for growth (b>1), it never crosses as x increases — it just keeps rising. Ext.
Spot the Error
  1. “The graph of y = x² + 4 doesn’t cross the x-axis, so the equation x² + 4 = 0 has two real solutions.”
  2. “To solve x² − 5x + 6 = 0 graphically, I read off the y-intercept.”
  3. “The graph of y = 5ˣ is a straight line.”
Show answers
  1. If the graph never crosses the x-axis, the equation has no real solutions at all — not two.
  2. To solve an equation graphically, you need the x-intercepts (roots), not the y-intercept — these give completely different information.
  3. y = 5ˣ is an exponential curve, not a straight line — only functions of the form y = mx + c are straight lines.

9. Mixed Practice — Bringing It All Together

Core & Extended
  1. Construct a table of values for y = x² − x − 2, for x from −2 to 3.
  2. Using your table, state the roots of x² − x − 2 = 0.
  3. For y = 6/x, find y when x = 1, 2, 3, 6.
Show answers
  1. x: −2,−1,0,1,2,3 → y: 4,0,−2,−2,0,4
  2. x = −1 and x = 2
  3. x=1: 6. x=2: 3. x=3: 2. x=6: 1
Extended-level mixed set Extended only
  1. For y = x³ − 4x, find y when x = −2, −1, 0, 1, 2.
  2. For y = 2ˣ + 1, find y when x = 0, 1, 2, 3.
  3. For y = 3/x², find y when x = 1, 3.
Show answers
  1. x=−2: 0. x=−1: 3. x=0: 0. x=1: −3. x=2: 0
  2. x=0: 2. x=1: 3. x=2: 5. x=3: 9
  3. x=1: 3. x=3: 1/3

Practice at Home

A large set again — take it steadily.

Part A — Tables of values and quadratics
  1. Construct a table of values for y = x² + 2x − 3, for x from −4 to 2.
  2. Using your table, state the roots of x² + 2x − 3 = 0.
  3. Construct a table of values for y = −x² + 4, for x from −3 to 3.
Part B — Reciprocal graphs
  1. For y = 8/x, find y when x = 1, 2, 4, 8.
  2. Explain why the graph of y = 8/x has two separate branches instead of one continuous curve.
Part C — Solving equations graphically
  1. A graph of y = x² − 3x − 4 crosses the x-axis at two points. Construct a table of values (x from −2 to 5) and state the roots.
  2. Describe how you would use a graph of y = x² − 3x − 4 to solve x² − 3x − 4 = 2x − 1.
Part D — Cubic and other power graphs Extended
  1. For y = x³ − x, find y when x = −2, −1, 0, 1, 2.
  2. For y = 5√x, find y when x = 0, 1, 4, 9.
  3. For y = 6/x², find y when x = 1, 2, 3.
Part E — Exponential graphs Extended
  1. For y = 4ˣ, find y when x = 0, 1, 2.
  2. A radioactive sample decays according to M = 80 × (0.9)ᵗ, where t is time in years. Find M when t = 0, 1, 2, 3 (round to 1 d.p.).
  3. Does y = (1/3)ˣ represent growth or decay? Explain how you know.
Part F — Applied / challenge
  1. A ball’s height is modelled by h = −x² + 6x, where x is horizontal distance in metres. Construct a table of values for x from 0 to 6, and find the two values of x where the ball is at ground level (h = 0).
  2. Sketch (in words) how the graphs of y = x² and y = x³ differ for negative values of x, and explain why.
  3. A bacteria colony doubles every hour, starting at 50. Write an exponential formula for the population P after t hours, and find the population after 5 hours.

Key Vocabulary Recap

Parabola — the U-shaped (or ∩-shaped) curve of a quadratic graph
Root — a value of x where a graph crosses the x-axis (y = 0)
Asymptote — a line a curve approaches but never touches
Turning point — a local maximum or minimum on a curve
Exponential growth/decay — a curve that increases/decreases faster and faster, based on a repeated multiplier Ext.

Syllabus Reference

This note covers Cambridge IGCSE Mathematics (0580) section C2.10 (Core subject content) and section E2.10 (Extended subject content) — Graphs of functions. Core is limited to linear (ax+b), quadratic (±x²+ax+b), and reciprocal (a/x) graphs with integer coefficients, plus solving associated equations graphically. Extended covers a much wider family of functions (axⁿ for various powers n, combined in sums of up to three terms) plus exponential graphs abˣ+c, marked “Extended only” above.

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