IGCSE Maths · Algebra and Graphs · Core & Extended

Sketching Curves: Key Features Without Plotting Every Point

A sketch is different from a plot — instead of calculating dozens of points, you identify the handful of features that define the shape (intercepts, symmetry, turning points, asymptotes) and draw a representative curve straight from those. Core sketches linear and quadratic graphs, and does not need turning points. Extended adds cubic, reciprocal, and exponential sketches, and does require turning points — tagged throughout.

1. Sketching vs Plotting

A plot (from the previous note) uses a table of values and precise coordinates, joined carefully. A sketch is faster and less exact — it should still be recognisably the right shape, correctly labelled with intercepts and any other required features, but doesn’t need graph paper or a ruler for the curve itself (a ruler is still needed for any straight lines or axes).

2. Sketching Linear Graphs

For a straight line, you only need two features: the y-intercept, and either the gradient or a second point (often the x-intercept).

Worked example Sketch y = 2x − 3.
y-intercept: x = 0 → y = −3, so the line passes through (0, −3).
x-intercept: y = 0 → 0 = 2x − 3 → x = 1.5, so the line passes through (1.5, 0).
Gradient is positive (2), so the line rises left to right.
Try it — Sketching Linear Graphs
  1. State the y-intercept and x-intercept you’d use to sketch y = 3x + 6.
  2. State the y-intercept and x-intercept you’d use to sketch y = −x + 4.
Show answers
  1. y-intercept (0, 6); x-intercept: 0 = 3x+6 → x = −2, so (−2, 0)
  2. y-intercept (0, 4); x-intercept: 0 = −x+4 → x = 4, so (4, 0)

3. Sketching Quadratic Graphs

For Core, a quadratic sketch needs: the roots (from factorising), the y-intercept, the overall shape (U or ∩, from the sign of the x² coefficient), and the line of symmetry — which sits exactly halfway between the two roots. Turning points are not required at Core level.

Worked example — Core level 2 3 line of symmetry Sketch y = x² − 5x + 6.
Factorise to find roots: (x − 2)(x − 3) = 0 → x = 2 and x = 3
y-intercept: x = 0 → y = 6
Shape: U-shaped (positive x² coefficient)
Line of symmetry: halfway between the roots, at x = 2.5 — you don’t need the minimum y-value itself at Core level, just the position of the symmetry line.
Try it — Sketching Quadratics (Core)
  1. Find the roots and y-intercept you’d use to sketch y = x² − 7x + 10.
  2. Find the roots you’d use to sketch y = x² − 9, and state the line of symmetry.
Show answers
  1. (x−2)(x−5)=0 → roots x=2,5; y-intercept (0,10)
  2. (x−3)(x+3)=0 → roots x=−3,3; line of symmetry x=0

4. Finding the Turning Point by Completing the Square Extended only

Extended sketches need the exact turning point. Writing the quadratic in completed square form a(x + p)² + q gives it directly: the turning point is at (−p, q).

Worked example 1 5 (3, −4) Sketch y = x² − 6x + 5, showing the turning point.
Roots: factorise → (x − 1)(x − 5) = 0 → x = 1 and x = 5
Complete the square: x² − 6x + 5 = (x − 3)² − 9 + 5 = (x − 3)² − 4
Turning point: (3, −4) (a minimum, since the x² coefficient is positive)
Try it — Turning Points Extended
  1. Find the turning point of y = x² − 4x + 1 by completing the square.
  2. Find the turning point of y = x² + 2x − 5 by completing the square, and state whether it’s a maximum or minimum.
Show answers
  1. (x−2)²−4+1=(x−2)²−3. Turning point: (2, −3)
  2. (x+1)²−1−5=(x+1)²−6. Turning point: (−1, −6), a minimum (positive x² coefficient).

5. Sketching Cubic Graphs Extended only

Factorise to find the roots, check the y-intercept, and use the sign of the x³ coefficient to determine the overall direction: positive means the curve rises from bottom-left to top-right; negative means the reverse.

Worked example −2 0 2 Sketch y = x³ − 4x.
Factorise: x³ − 4x = x(x² − 4) = x(x − 2)(x + 2)
Roots: x = −2, 0, 2
y-intercept: 0 (matches the root at x = 0)
The x³ coefficient is positive, so the curve rises overall from bottom-left to top-right, with the characteristic “wiggle” between the roots.
Worked example — a simpler cubic with no turning points Sketch y = x³ − 8.
Root: x³ = 8 → x = 2 (only one real root)
y-intercept: −8
Since there’s no x² or x term, this cubic has no turning points at all — it’s the graph of y = x³ shifted down by 8, a single smooth rising curve.
Try it — Sketching Cubics Extended
  1. Find the roots you’d use to sketch y = x³ − 9x.
  2. Find the root of y = x³ + 27, and describe the overall shape.
Show answers
  1. x(x²−9)=x(x−3)(x+3)=0 → roots x=−3, 0, 3
  2. x³=−27 → x=−3. Since there’s no x² or x term, it’s a smooth rising curve (y=x³ shifted up by 27), with no turning points.

6. Sketching Reciprocal Graphs Extended only

For y = m/x + n, the asymptotes shift away from the axes: the vertical asymptote stays at x = 0, but the horizontal asymptote moves to y = n (since m/x → 0 as x gets very large, leaving just y = n).

Worked example y = 1 (asymptote) Sketch y = 2/x + 1.
Vertical asymptote: x = 0 (the function is undefined there)
Horizontal asymptote: y = 1 (shifted up from y = 0, because of the “+1”)
x-intercept: 0 = 2/x + 1 → 2/x = −1 → x = −2
Try it — Sketching Reciprocals Extended
  1. State the horizontal and vertical asymptotes of y = 4/x − 2.
  2. Find the x-intercept of y = 4/x − 2.
Show answers
  1. Vertical asymptote x = 0; horizontal asymptote y = −2
  2. 0 = 4/x − 2 → 4/x = 2 → x = 2

7. Sketching Exponential Graphs Extended only

For y = mˣ + n, the horizontal asymptote is at y = n (shifted from y = 0), and the y-intercept is found by substituting x = 0.

Worked example Sketch y = 3ˣ − 2.
Horizontal asymptote: y = −2 (shifted down from y = 0)
y-intercept: x = 0 → y = 3⁰ − 2 = 1 − 2 = −1
Shape: standard exponential growth curve, shifted down by 2, approaching but never reaching y = −2 as x decreases.
Try it — Sketching Exponentials Extended
  1. State the horizontal asymptote and y-intercept of y = 2ˣ + 3.
  2. State the horizontal asymptote and y-intercept of y = 5ˣ − 4.
Show answers
  1. Asymptote y = 3; y-intercept: 2⁰+3 = 4
  2. Asymptote y = −4; y-intercept: 5⁰−4 = 1−4 = −3
Common mistakes to watch for
  • Drawing a quadratic sketch with the line of symmetry in the wrong place — it must sit exactly halfway between the two roots.
  • Forgetting turning points are required for Extended sketches but not Core ones — check which tier a question is aimed at.
  • Placing a horizontal asymptote at y = 0 out of habit, instead of checking for a “+n” or “−n” shift.
  • Missing a root when factorising a cubic — always check for a common factor first (like a factor of x), then factorise what’s left.
  • Drawing a cubic with only one turning point when it should have two (or vice versa) — check whether the cubic has an x² or x term at all; y = ax³ + d alone has no turning points.

8. Quick-Fire Challenge Round

True or False?
  1. Core-level quadratic sketches need the exact turning point.
  2. The line of symmetry of a quadratic sits halfway between its two roots.
  3. A cubic graph can have at most two turning points.
  4. Shifting y = 1/x to y = 1/x + 3 moves the horizontal asymptote to y = 3.
Show answers
  1. False — Core needs roots, y-intercept, shape, and line of symmetry, but not the turning point itself.
  2. True.
  3. True. Ext.
  4. True. Ext.
Spot the Error
  1. “y = x² − 6x + 8 has roots x = 2 and x = 4, so the line of symmetry is x = 3.” (is this actually an error?)
  2. “To find the turning point of y = x² + 4x + 1, I completed the square to get (x+2)² − 3, so the turning point is (2, −3).”
  3. “y = 2ˣ − 5 has a horizontal asymptote at y = 2.”
Show answers
  1. This one is actually correct — (2+4)/2 = 3, so x = 3 is indeed the line of symmetry. Not every “spot the error” has an error; always check rather than assuming.
  2. The turning point is (−p, q) when the completed square form is (x+p)² + q — since p = 2, the turning point is (−2, −3), not (2, −3).
  3. The asymptote is at y = −5 (matching the “−5” shift), not y = 2.

9. Mixed Practice — Bringing It All Together

Core & Extended
  1. State the y-intercept and x-intercept for sketching y = 4x − 8.
  2. Find the roots and y-intercept for sketching y = x² − x − 6.
  3. State the line of symmetry for y = x² − x − 6.
Show answers
  1. y-intercept (0,−8); x-intercept (2,0)
  2. (x−3)(x+2)=0 → roots x=3,−2; y-intercept (0,−6)
  3. Halfway between −2 and 3: x = 0.5
Extended-level mixed set Extended only
  1. Find the turning point of y = x² − x − 6 by completing the square.
  2. Find the roots of y = x³ − 16x for sketching.
  3. State the asymptotes of y = 3/x + 4.
Show answers
  1. (x−0.5)²−0.25−6=(x−0.5)²−6.25. Turning point: (0.5, −6.25)
  2. x(x²−16)=x(x−4)(x+4)=0 → roots x=0,4,−4
  3. Vertical asymptote x=0; horizontal asymptote y=4

Practice at Home

A large set again — take it steadily.

Part A — Linear and quadratic sketches (Core & Extended)
  1. State the y-intercept and x-intercept for sketching y = 5x + 10.
  2. Find the roots and y-intercept for sketching y = x² − 8x + 12.
  3. State the line of symmetry for y = x² − 8x + 12.
  4. Find the roots for sketching y = x² − 16.
Part B — Turning points Extended
  1. Find the turning point of y = x² − 8x + 12 by completing the square.
  2. Find the turning point of y = x² + 6x + 4 by completing the square.
  3. Find the turning point of y = x² − 2x − 8, and state whether it’s a maximum or minimum.
Part C — Cubic sketches Extended
  1. Find the roots for sketching y = x³ − x.
  2. Find the roots for sketching y = x³ − 25x.
  3. Find the root of y = x³ + 1, and describe whether it has turning points.
Part D — Reciprocal and exponential sketches Extended
  1. State the asymptotes of y = 5/x − 3.
  2. Find the x-intercept of y = 5/x − 3.
  3. State the asymptote and y-intercept of y = 4ˣ − 1.
Part E — Applied / challenge
  1. A quadratic has roots x = −4 and x = 6, and passes through (0, −24). Find its equation in the form y = x² + bx + c, then find its turning point.
  2. Sketch (in words) how the graph of y = x³ − 3x² compares to y = x³ − 3x — specifically, describe how you’d find each one’s roots and what that tells you about how many times each crosses the x-axis.
  3. Explain why the graph of y = 1/x² never dips below the x-axis, using the fact that x² is always positive (for x ≠ 0).

Key Vocabulary Recap

Sketch — a graph showing key features accurately, without a full table of values
Line of symmetry — the vertical line a parabola is symmetrical about
Turning point — a maximum or minimum point on a curve Ext.
Vertical/horizontal asymptote — a line a curve approaches but never crosses Ext.

Syllabus Reference

This note covers Cambridge IGCSE Mathematics (0580) section C2.11 (Core subject content) and section E2.11 (Extended subject content) — Sketching curves. Core covers linear and quadratic sketches, requiring symmetry and roots but not turning points. Extended adds cubic, reciprocal, and exponential sketches, and requires turning points (including finding them by completing the square), asymptotes, and full symmetry — all marked “Extended only” above.

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