Sketching curves.
Show the features that identify a curve, including turning points, symmetry and asymptotes.
Sketch linear, quadratic, cubic, reciprocal and exponential curves · label key features
A sketch shows the defining features
A sketch communicates a graph’s important features without needing a full accurately scaled table. Recognising the function family gives you a starting shape before you calculate key points.
Look at the highest power, the sign of its coefficient and any denominator involving x. These clues suggest a parabola, cubic or reciprocal-type curve. Then locate intercepts and other features to anchor the sketch rather than drawing a remembered shape in an arbitrary position.
Label roots, intercepts, turning points and asymptotes where relevant. A sketch need not be a table-of-values plot, but its shape and key coordinates must be clear.
U-shaped parabola. Roots −1 and 3; turning point (1, −4).
| x | -2 | -1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y | 5 | 0 | -3 | -4 | -3 |
| Family | Typical form | Features to show |
|---|---|---|
| Linear | ax + by = c | Straight line; intercepts (vertical lines are possible). |
| Quadratic | y = ax² + bx + c | Parabola, turning point, symmetry, roots. |
| Cubic | y = ax³ + b or ax³ + bx² + cx | End behaviour, intercepts; may have turning points. |
| Reciprocal | y = ax + b | Two branches; asymptotes x = 0 and y = b. |
| Exponential | y = arˣ + b, r > 0 | For a ≠ 0, r ≠ 1: asymptote y = b; y-intercept a + b. |
Pause and explainWhat does a positive coefficient of x² tell you about a quadratic?
Its parabola opens upward. It does not by itself tell you the intercepts or where its lowest point lies.
Complete the square to locate a turning point
The turning point of a parabola is where it changes from decreasing to increasing, or the reverse. Its completed-square form shows how the basic y = x² shape has been shifted.
In y = (x − a)² + b, the square is smallest at x = a, so the turning point is (a, b) for an upward-opening parabola. Notice the sign inside the bracket: x − 3 becomes zero at positive 3.
Sketch y = 2x² − 8x + 6.
Turning point: (2, −2), a minimum. Symmetry line: x = 2. Roots: x = 1 and 3. y-intercept: (0, 6).
For y = a(x − h)² + k, the turning point is (h, k). Positive a gives a minimum; negative a gives a maximum.
Pause and explainWhere is the turning point of y = (x − 3)² + 2?
(3, 2). At x = 3 the square is zero, leaving the smallest output 2.
Cubic shape and asymptotes
A cubic can cross the x-axis more than once and can change direction twice. A reciprocal-type graph may instead have a line that it approaches without meeting in the relevant branch.
Such an approached line is an asymptote. Identify where a denominator becomes zero to find a possible vertical asymptote, and examine the function’s form for horizontal behaviour. Do not join branches through a forbidden x-value or treat an asymptote as an ordinary part of the curve.
The graph y = ax³ + b has rotational symmetry about (0, b). For non-zero a it has a stationary point of inflection there, not a maximum or minimum. Cubics with extra terms can have two turning points, as in the example below.
Sketch y = x³ − 4x.
Factor: y = x(x − 2)(x + 2), so roots are −2, 0, 2. The graph has origin symmetry; with a positive leading coefficient it falls to the left and rises to the right.
Sketch y = 2x + 1.
Vertical asymptote: x = 0. Horizontal asymptote: y = 1. Root: x = −2. There is no y-intercept because x = 0 is excluded.
Pause and explainWhy must y = 1x have separate branches around x = 0?
The expression is undefined at x = 0, so the graph cannot contain a point there or pass continuously through it.
Your turn. Label before you draw.
List the features that your sketch must show: shape, intercepts, turning points and any asymptotes. Calculate those features before joining the curve.
Label important coordinates explicitly and keep branches on the correct sides of asymptotes. A convincing sketch explains the equation’s structure, even when the spacing is not drawn to exact scale.
Give the turning point, symmetry line and roots of y = (x + 1)² − 9.
Reveal worked answer
State the asymptotes of y = −3x + 4.
Reveal worked answer
For y = 3 × 2ˣ − 5, state the y-intercept and horizontal asymptote.