A sketch is not a plot. You do not need a table of values, just the right shape and the points that matter: where the curve crosses the axes, where it turns, and the lines it approaches.
Sketch = correct shape + labelled key points. Set \(x = 0\) for the \(y\)-intercept and \(y = 0\) for the roots; for a quadratic the line of symmetry is halfway between the roots, and completing the square gives the turning point.
Fig. 1: four sketches with their key points: (a) roots 1 and 5, minimum (3, −4); (b) roots −3, 0 and 2; (c) asymptotes x = 0 and y = 1; (d) asymptote y = −2, through (0, −1) and (1, 0)
Section 01
Straight lines and parabolas.
In the lab's sketching tab, change the coefficients and watch the key features move: the intercepts, the line of symmetry and the turning point. Notice that a negative \(x^2\) term always turns the parabola upside down.
Lines \(ax + by = c\): put \(x = 0\) to find where it meets the \(y\)-axis and \(y = 0\) for the \(x\)-axis, then join the two points.
Quadratics \(y = ax^2 + bx + c\): U-shaped if \(a > 0\), ∩-shaped if \(a < 0\). The \(y\)-intercept is \(c\).
Roots: factorise and set each bracket to zero.
Symmetry: the line of symmetry is the vertical line halfway between the roots.
Worked example Sketching a straight line
Sketch \(2x + 3y = 12\).
\(x = 0\): \(3y = 12\), so \(y = 4\). The line meets the \(y\)-axis at \((0, 4)\).
\(y = 0\): \(2x = 12\), so \(x = 6\). It meets the \(x\)-axis at \((6, 0)\).
Join the two points with a ruled line: it slopes down from left to right.
Answer: A straight line through \((0, 4)\) and \((6, 0)\)
Worked example Sketching a quadratic
Sketch \(y = x^2 - 6x + 5\), showing the roots, the \(y\)-intercept and the line of symmetry.
\((x - 3)^2\) is never negative, so the smallest value of \(y\) is \(-4\), when \(x = 3\).
The curve is U-shaped, so this is a minimum.
Answer: Minimum point \((3, -4)\)
Your turn
Sketch \(y = -x^2 + 2x + 8\). Show the roots, the \(y\)-intercept and the turning point.
\(-x^2 + 2x + 8 = -(x^2 - 2x - 8) = -(x - 4)(x + 2)\), so roots \(x = 4\) and \(x = -2\). \(y\)-intercept 8. Symmetry line \(x = 1\), and \(y = -1 + 2 + 8 = 9\) there. ∩-shaped, maximum \((1, 9)\).
Examiner trap
In \((x - 3)^2 - 4\) the turning point is \((3, -4)\), not \((-3, -4)\). The sign inside the bracket flips; the number outside does not.
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
Where does \(y = x^2 - 6x + 5\) cross the \(y\)-axis?
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What is the line of symmetry of \(y = (x - 1)(x - 5)\)?
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Where does \(2x + 3y = 12\) cross the \(y\)-axis?
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(Extended) What is the horizontal asymptote of \(y = \dfrac{3}{x} - 2\)?
Section 02
Cubic curves.
Extended. A cubic has an \(x^3\) term. With a positive \(x^3\) term it starts bottom-left and ends top-right; a negative one starts top-left and ends bottom-right.
\(y = ax^3 + b\) has the shape of \(y = x^3\) (stretched if \(a \ne 1\)), moved up by \(b\): it crosses the \(x\)-axis once.
\(y = ax^3 + bx^2 + cx\) always passes through the origin, because every term contains \(x\). Take out the factor \(x\) to find the other roots.
Worked example Roots of a cubic (Extended)
Sketch \(y = x^3 + x^2 - 6x\).
Take out \(x\): \(y = x(x^2 + x - 6)\).
Factorise the quadratic: \(y = x(x + 3)(x - 2)\).
Roots: \(x = 0, -3, 2\).
Positive \(x^3\): start bottom-left, rise through \(-3\), turn, fall through 0, turn, rise through 2.
Answer: An S-shaped curve through \((-3, 0)\), \((0, 0)\) and \((2, 0)\) (Fig. 1b)
Worked example A falling cubic (Extended)
Sketch \(y = 8 - x^3\).
\(y\)-intercept: \(x = 0\) gives \(y = 8\).
Root: \(8 - x^3 = 0\), so \(x^3 = 8\) and \(x = 2\).
The \(x^3\) term is negative, so the curve falls from top-left to bottom-right, flattening briefly at \((0, 8)\).
Answer: A falling cubic through \((0, 8)\) and \((2, 0)\)
Examiner trap
Do not forget the root \(x = 0\) when you take out a factor of \(x\). \(x^3 + x^2 - 6x = 0\) has THREE solutions, not two.
Section 03
Reciprocals, exponentials and asymptotes.
Extended. An asymptote is a line that a curve gets closer and closer to but never reaches. Draw it as a dashed line and label its equation.
Graph
Asymptotes
Key points
\(y = \dfrac{a}{x} + b\)
\(x = 0\) and \(y = b\)
no \(y\)-intercept; crosses the \(x\)-axis where \(\dfrac{a}{x} = -b\)
\(y = a r^{x} + b\)
\(y = b\) only
\(y\)-intercept \(a + b\), because \(r^0 = 1\)
Worked example A reciprocal graph (Extended)
Sketch \(y = \dfrac{2}{x} + 1\), showing its asymptotes and any intercepts.
\(\dfrac{2}{x}\) cannot be worked out at \(x = 0\), so \(x = 0\) is a vertical asymptote.
As \(x\) gets very large, \(\dfrac{2}{x}\) gets close to 0, so \(y\) gets close to 1: \(y = 1\) is a horizontal asymptote.
\(x\)-intercept: \(\dfrac{2}{x} + 1 = 0\), so \(\dfrac{2}{x} = -1\) and \(x = -2\).
No \(y\)-intercept, since \(x = 0\) is not allowed.
Answer: Two branches, asymptotes \(x = 0\) and \(y = 1\), crossing the \(x\)-axis at \((-2, 0)\) (Fig. 1c)
Worked example An exponential graph (Extended)
Sketch \(y = 2^{x} - 2\).
\(y\)-intercept: \(2^0 - 2 = 1 - 2 = -1\).
\(x\)-intercept: \(2^x = 2\), so \(x = 1\).
As \(x\) becomes very negative, \(2^x\) gets close to 0, so \(y\) gets close to \(-2\): asymptote \(y = -2\).
As \(x\) increases, \(2^x\) grows faster and faster.
Answer: Rising curve through \((0, -1)\) and \((1, 0)\), with asymptote \(y = -2\) on the left (Fig. 1d)
Examiner trap
\(y = 2^x\) does NOT pass through the origin: \(2^0 = 1\). And an exponential curve never crosses its asymptote, so your sketch must not touch the dashed line.
Stage 2 · Exam
Exam-style questions
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(Extended) By completing the square, find the turning point of \(y = x^2 + 8x + 11\).
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Which statement is true for the graph of \(y = (x + 2)(x - 6)\)?
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(Extended) Which describes the graph of \(y = 2^{x} - 2\)?
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(Extended) A cubic curve crosses the \(x\)-axis at \(-3\), 0 and 2, rising from bottom-left to top-right. What is its equation?
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