This note is about drawing and reading graphs of functions: what a good sketch must show, how to recognise common graph shapes, and how to sketch a graph from a description or a real-life situation. Sketching lets you see a problem before you solve it, and it is a skill used in nearly every topic.
Key ideas
The graph of a function is the set of points \((x, f(x))\) drawn on axes, written as the equation \(y = f(x)\). A sketch is different from an accurate plot. It does not need a scale on every grid line, but it must show the right shape and the important numbers.
A good sketch includes:
- Labelled axes. In a context, use the real quantities, such as time \(t\) in seconds and height \(h\) in metres.
- The correct overall shape and direction, including what happens at each end.
- Where the graph crosses the axes. For the \(y\)-intercept, put \(x = 0\). For the \(x\)-intercepts, solve \(f(x) = 0\).
- Turning points (maximum and minimum points) when the shape has them.
- Asymptotes, drawn as dashed lines and labelled with their equations.
- Any coordinates given in the question, written next to the points.
Recognising shapes. Knowing the standard shapes helps you sketch quickly.
- Linear, \(y = mx + c\): a straight line.
- Quadratic, \(y = ax^2 + bx + c\): a parabola, opening up if \(a \gt 0\) and down if \(a \lt 0\).
- Cubic: an S-like curve with up to two turning points. A positive leading coefficient rises to the right.
- Reciprocal, \(y = \dfrac{1}{x}\): two separate branches with the axes as asymptotes.
- Exponential, \(y = a^x\): a curve through \((0, 1)\) that approaches the \(x\)-axis on one side.
- Sine and cosine: repeating waves between a minimum and a maximum value.
Sketching from a context. Many questions describe a situation. Translate each phrase into a feature of the graph.
- "Starts at" tells you the \(y\)-intercept.
- "Reaches its greatest value" tells you the maximum point.
- "Levels off" or "approaches" tells you a horizontal asymptote.
- "Hits the ground" means the height is 0, so it is an \(x\)-intercept.
Also think about the domain. A graph of a physical quantity often stops at a certain time, so draw it only for the values that make sense.
Using technology. You can use a GDC to draw a graph and then copy the shape onto paper. Copy the window you used, label the axes, and mark the key points with their coordinates. You can also use a GDC to graph the sum or difference of two functions, such as \(y = f(x) + g(x)\), by entering the combined expression.
Definitions and formulae
| Term or formula | Meaning / when to use |
|---|---|
| \(y\)-intercept | Where the graph crosses the \(y\)-axis. Set \(x = 0\). |
| \(x\)-intercept (zero, root) | Where the graph crosses the \(x\)-axis. Solve \(f(x) = 0\). |
| Turning point | A point where the graph changes from rising to falling, or the reverse. |
| Asymptote | A line the graph gets closer and closer to without reaching, drawn dashed. |
| \(y = a(x - p)(x - q)\) | Quadratic with \(x\)-intercepts at \(p\) and \(q\). Useful for sketching. |
| End behaviour | What happens to \(y\) as \(x\) becomes very large positive or very large negative. |
| Sketch | A diagram showing shape and key features, not an accurately scaled plot. |
Worked examples
Example 1 Sketch the graph of \(y = (x + 1)(x - 2)(x - 4)\), showing all intercepts.
Step 1: \(x\)-intercepts. The expression is zero when \(x + 1 = 0\), \(x - 2 = 0\) or \(x - 4 = 0\). So the graph crosses the \(x\)-axis at \(x = -1\), \(x = 2\) and \(x = 4\).
Step 2: \(y\)-intercept. Put \(x = 0\): \(y = (1)(-2)(-4) = 8\). The graph passes through \((0, 8)\).
Step 3: shape. When multiplied out, the highest power is \(x^3\) with coefficient 1, which is positive. So the graph rises towards the top right and falls towards the bottom left.
Step 4: draw. Start at the bottom left, cross the axis at \(-1\), rise through \((0, 8)\) to a maximum between \(x = -1\) and \(x = 2\), come back down through \(x = 2\), reach a minimum between 2 and 4, and then rise through \(x = 4\).
Answer: a cubic crossing the \(x\)-axis at \(-1\), \(2\), \(4\) and the \(y\)-axis at \(8\), with a maximum then a minimum.
Example 2 A football is kicked from a height of 1.5 m. Its height in metres after \(t\) seconds is \(h = -5t^2 + 20t + 1.5\). Sketch \(h\) against \(t\) until the ball hits the ground, and give the key values.
Step 1: starting point. At \(t = 0\), \(h = 1.5\). The graph starts at \((0, 1.5)\).
Step 2: shape. The coefficient of \(t^2\) is negative, so the graph is an upside-down parabola.
Step 3: the highest point. The axis of symmetry is at \(t = -\dfrac{b}{2a} = -\dfrac{20}{2(-5)} = 2\). Then \(h(2) = -20 + 40 + 1.5 = 21.5\). The maximum is at \((2, 21.5)\).
Step 4: when it lands. Solve \(-5t^2 + 20t + 1.5 = 0\). Using the quadratic formula (or the polynomial solver on your GDC), \(t = \dfrac{20 \pm \sqrt{430}}{10}\). The positive value is \(t = 4.07\) s to 3 s.f. The negative value is outside the domain, because time cannot be negative.
Step 5: draw from \((0, 1.5)\), up to \((2, 21.5)\), and down to \((4.07, 0)\). Stop there. Label the axes with \(t\) (s) and \(h\) (m).
Answer: maximum height 21.5 m at \(t = 2\) s; the ball lands at \(t = 4.07\) s.
Example 3 A mug of tea cools in a room at \(20^\circ\)C. Its temperature in \(^\circ\)C after \(t\) minutes is \(T = 20 + 60(0.9)^t\), for \(t \geq 0\). Sketch the graph and describe what happens for large \(t\).
Step 1: starting value. At \(t = 0\), \((0.9)^0 = 1\), so \(T = 80\). The graph starts at \((0, 80)\).
Step 2: some values. At \(t = 10\), \((0.9)^{10} = 0.3487\), so \(T = 20 + 60(0.3487) = 40.9\). At \(t = 20\), \((0.9)^{20} = 0.1216\), so \(T = 27.3\).
Step 3: long-term behaviour. As \(t\) grows, \((0.9)^t\) gets closer to 0, so \(T\) gets closer to 20 but never reaches it. The line \(T = 20\) is a horizontal asymptote.
Step 4: draw a decreasing curve from \((0, 80)\) that flattens out towards the dashed line \(T = 20\). It is steep at first and then gentler.
Answer: the tea starts at \(80^\circ\)C and cools towards the room temperature of \(20^\circ\)C, never going below it.
Common errors
- Common error: drawing a graph that "bounces off" the axis or ends abruptly. Unless the domain is restricted, curves continue beyond the edges of your sketch, so show arrows or extend the curve.
- Common error: letting a curve cross its own asymptote or touch it. Draw the curve getting close and then flattening along it.
- Common error: leaving out the axis labels or the coordinates of key points. A sketch without them is incomplete.
- Common error: ignoring the domain in a context. Do not draw negative times or negative lengths.
- Common error: forgetting that the \(y\)-intercept needs \(x = 0\) and the \(x\)-intercepts need \(y = 0\), and swapping the two.
Exam tip: when the question says "sketch", give shape, intercepts and any turning points or asymptotes with their values. When it says "draw", use a ruler and plot accurately. Do not spend time plotting many points for a sketch.
Check yourself
- State the \(y\)-intercept and \(x\)-intercepts of \(y = (x - 3)(x + 5)\).
Answer
When \(x = 0\), \(y = (-3)(5) = -15\). The \(x\)-intercepts are at \(x = 3\) and \(x = -5\). - A graph starts at \((0, 5)\), decreases, and gets closer to the line \(y = 1\) without reaching it. State the equation of its horizontal asymptote and say whether the graph could have an \(x\)-intercept.
Answer
The asymptote is \(y = 1\). The graph stays above 1 (it starts at 5 and decreases towards 1), so it never reaches \(y = 0\). There is no \(x\)-intercept. - For \(y = -2x^2 + 8x\), find the \(x\)-intercepts and the coordinates of the maximum point.
Answer
Factorise: \(y = -2x(x - 4)\), so the intercepts are \(x = 0\) and \(x = 4\). The maximum is halfway at \(x = 2\), where \(y = -8 + 16 = 8\). The maximum point is \((2, 8)\). - Describe the end behaviour of the cubic \(y = -x^3 + 2x\).
Answer
The leading coefficient is negative. So for large positive \(x\), \(y\) becomes very negative, and for large negative \(x\), \(y\) becomes very positive. The graph falls from top left to bottom right.