Maths · 3 Functions

3.2 Sketching graphs on a calculator

Before you start

The standard shapes of linear, quadratic, cubic, reciprocal, exponential and trigonometric graphs (3.1).
Substituting a value into a function.
That a zero of a function is where its graph crosses the $x$ -axis.

Syllabus objectives covered (1)

Use a graphic display calculator to: (a) sketch the graph of a function (b) produce a table of values (c) plot points (d) find zeros, local maxima or local minima (e) find the intersection of the graphs of functions (f) find the vertex of a quadratic.

Part 1 of 4

This subtopic is calculator papers only

Before anything else: 3.2 is not examined in Paper 2. The syllabus lists it among the subtopics assessed in the calculator papers only.
That is not a loophole to exploit but a signal about what is being tested. The skill here is using the calculator well, not reproducing its work by hand.
So when you practise this subtopic, have the graphic display calculator in front of you. Reading about it is close to useless.
The syllabus notes add something important: questions may use unfamiliar functions not mentioned anywhere else in the syllabus. You are not expected to recognise them — you are expected to be able to graph anything you are handed.

Part 2 of 4

Getting the window right

The single most common problem is a blank or useless screen, and it is almost always the window, not the function.
Enter the function, then set the viewing window: the minimum and maximum for $x$ and for $y$ .
If nothing appears, the interesting part of the graph is outside your window. Zoom out substantially — try $- 10$ to $10$ on both axes first, then wider.
If the graph looks like a vertical line or a flat line, the scales are badly matched. Adjust the $y$ -range to suit the values the function actually takes.
A quick way to choose the window: use the table of values first. Generate $f (x)$ for a spread of $x$ values, see what range the outputs cover, and set the window to match.
For trigonometric graphs, check the calculator is in degrees, not radians. This syllabus works in degrees throughout, and a graph in the wrong mode looks completely wrong.
A sensible window for $y = \sin x$ is $0 ° \leq x \leq 360 °$ and $- 1.5 \leq y \leq 1.5$ , which fills the screen with exactly one cycle.

Part 3 of 4

The six things it will be asked to do

Sketch the graph. Copy the calculator's display onto your paper, showing the overall shape and any key points, and label the axes with their scales.
A sketch is not a plot. It should show the shape, the intercepts and the turning points correctly, but need not be to scale.
Produce a table of values. Use the table function, setting a start value and a step. This is useful for checking a graph, filling in a table in a question, or choosing a window.
Plot points. Enter a list of coordinates and display them, which is how you would check whether data lies on a particular curve.
Find zeros. The zeros are where the curve crosses the $x$ -axis. Use the calculator's root or zero function, and repeat for each crossing — it finds one at a time.
Find local maxima and minima. These are the turning points. 'Local' matters: a cubic can have a local maximum that is not the highest point on the whole graph.
Find intersections. Enter both functions and use the intersection tool. This is how you solve any equation you cannot solve algebraically — put one side in as $Y_{1}$ , the other as $Y_{2}$ , and find where they meet.
Find the vertex of a quadratic. The vertex is its single turning point — a minimum if the parabola opens upwards, a maximum if downwards. Use the same maximum/minimum tool.

Part 4 of 4

Writing down what the calculator tells you

A calculator answer with no working is still an answer, but a bare number often loses marks it need not.
Say what you did and what you found: 'intersection of $y = x^{3}$ and $y = 2 x + 1$ at $x = 1.62$ '. That sentence earns method marks even if the value is slightly out.
Give non-exact answers to 3 significant figures, or 1 decimal place for angles in degrees, unless the question specifies otherwise.
Do not round partway through. If an answer feeds into a later part, use the calculator's unrounded value, not the rounded one you wrote down.
Look for every solution. The calculator finds one intersection or one zero at a time, near wherever you point it. A cubic can cross the axis three times.
So before you start hunting, look at the whole graph and count how many crossings or intersections there are. Then find that many.
Finally, sanity-check against the shape. If you know from 3.1 that a parabola has at most two roots, three answers means something has gone wrong.

Worked examples

Example 1. Use a graphic display calculator to solve $x^{3} - 4 x + 1 = 0$ , giving all solutions correct to 3 significant figures.

Show the working

Enter $Y_{1} = x^{3} - 4 x + 1$ and display it with a window of about $- 4 \leq x \leq 4$ and $- 6 \leq y \leq 6$ .
Look at the whole curve first and count the crossings. It is a cubic and it crosses the $x$ -axis three times, so expect three answers.
Use the zero function three times, pointing the calculator near each crossing in turn.
The three roots are approximately $- 2.11$ , $0.254$ and $1.86$ .
Check one by substitution: ${(0.254)}^{3} - 4 (0.254) + 1 = 0.016 - 1.016 + 1 \approx 0$ . Close enough to confirm.

Answer: $x = - 2.11$ , $x = 0.254$ or $x = 1.86$ (3 s.f.).

Example 2. Use a graphic display calculator to find the vertex of $y = 2 x^{2} - 8 x + 3$ , and state whether it is a maximum or a minimum.

Show the working

Enter the function and display it. The coefficient of $x^{2}$ is positive, so the parabola opens upwards and the vertex will be a minimum.
Set a window that shows the turning point clearly — around $- 1 \leq x \leq 5$ and $- 6 \leq y \leq 6$ works.
Use the minimum function and let the calculator locate the turning point.
It returns $x = 2$ , $y = - 5$ .
Confirm by substitution: $2 (4) - 8 (2) + 3 = 8 - 16 + 3 = - 5$ . The vertex is exact, so no rounding is needed.

Answer: The vertex is at $(2, - 5)$ , and it is a minimum.

Example 3. The population of a colony is modelled by $P = 200 \times {1.15}^{t}$ , where $t$ is the time in years. Use a graphic display calculator to find, to the nearest year, when the population first exceeds $600$ .

Show the working

This is an intersection problem. Enter $Y_{1} = 200 \times {1.15}^{x}$ and $Y_{2} = 600$ .
Choose a window that will contain the crossing. Start with $0 \leq x \leq 20$ and $0 \leq y \leq 800$ .
Use the intersection function to find where the curve meets the horizontal line.
The calculator gives $x = 7.86$ .
Interpret it in context. The population exceeds 600 partway through year 8, so it first exceeds 600 after 8 years — round up, not to nearest, because at $t = 7$ it has not yet got there.

Answer: After $8$ years.

Where people go wrong

Mistake: Concluding a function has no solutions because the screen looks blank.

Instead: Zoom out and adjust the window — the graph is almost certainly off-screen.

The default window is small. A blank screen is a statement about the window, not about the function.

Mistake: Finding one zero and stopping.

Instead: Count the crossings on the whole graph, then find each one.

The calculator locates one root at a time, near where you point it. A cubic can have three.

Mistake: Leaving the calculator in radian mode for a trigonometric graph.

Instead: Set it to degrees.

This syllabus works in degrees throughout, and in radians a sine curve over $0$ to $360$ looks like an unrecognisable ripple.

Mistake: Writing down only the final number with no indication of method.

Instead: State what you graphed and what you found.

Method marks are available for the correct approach, and communication is separately assessed on Paper 6.

Mistake: Rounding a calculator value and using the rounded version in the next part.

Instead: Carry the full unrounded value forward.

The syllabus accuracy rules say so explicitly, and early rounding loses accuracy marks.

Check yourself

Your calculator screen appears blank after entering a function. What is the first thing to check?
Answer
The viewing window. Zoom out or set a wider range of $x$ and $y$ so the graph comes into view.
How would you use a graphic display calculator to solve $2^{x} = x + 5$ ?
Answer
Enter $Y_{1} = 2^{x}$ and $Y_{2} = x + 5$ , then use the intersection function to find every point where the two graphs meet.
What is the vertex of a quadratic, and how do you find it on a calculator?
Answer
Its single turning point. Use the minimum function if the parabola opens upwards, or the maximum function if it opens downwards.
Why does the syllabus say questions may involve unfamiliar functions?
Answer
Because the skill being tested is using the calculator competently, not recognising particular functions.
To what accuracy should a non-exact calculator answer be given?
Answer
3 significant figures, or 1 decimal place for angles in degrees, unless the question says otherwise.

Exam questions

Try each part first. Open the mark scheme after you have written your answer.

Question 1. $f (x) = x^{3} - 3 x^{2} + 1$

(a) Sketch the graph of $y = f (x)$ for $- 2 \leq x \leq 4$ , showing the turning points. 2 marks

Mark scheme

B1 correct cubic shape with two turning points
B1 maximum near (0, 1) and minimum near (2, -3) indicated

(b) Write down the coordinates of the local maximum and the local minimum. 2 marks

Mark scheme

B1 (0, 1)
B1 (2, -3)

(c) Solve $f (x) = 0$ , giving all solutions correct to 3 significant figures. 3 marks

Mark scheme

B1 -0.532 awrt
B1 0.653 awrt
B1 2.88 awrt

Question 2. The value of a car, $$ V$ , after $t$ years is modelled by $V = 18000 \times {0.82}^{t}$ .

(a) Work out the value of the car after $3$ years. 2 marks

Mark scheme

M1 18000 × 0.82³ seen oe
A1 9930 awrt (accept 9929.4…)

(b) Determine, using a graphic display calculator, the number of complete years after which the value first falls below $$ 5000$ . 3 marks

Mark scheme

M1 graphs V = 18000 × 0.82^t and V = 5000 and seeks the intersection oe
A1 t = 6.45 awrt seen
A1 7 (complete years); dep on a value between 6 and 7 seen

Question 3. Two functions are given: $f (x) = \frac{12}{x}$ and $g (x) = x + 1$ .

(a) Solve $f (x) = g (x)$ , giving your answers correct to 3 significant figures. 3 marks

Mark scheme

M1 both functions graphed and the intersections sought
A1 3.00 awrt
A1 -4.00 awrt

(b) Explain why the graphs meet in two places rather than one. 1 mark

Mark scheme

B1 the reciprocal graph has two separate branches, and the line crosses each one once oe

Words to know

Viewing window: The range of $x$ and $y$ values shown on the calculator screen.
Zero (root): A value of $x$ for which $f (x) = 0$ , shown by the graph crossing the $x$ -axis.
Local maximum / minimum: A turning point that is highest or lowest in its own neighbourhood, though not necessarily on the whole graph.
Vertex: The single turning point of a quadratic — a minimum if it opens upwards, a maximum if downwards.
Intersection: A point where two graphs meet, giving a solution of the equation formed by setting them equal.

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