A LevelMathsCambridge (CIE)Pure 1Exam QuestionsAlgebra & FunctionsGraphs of Functions

Graphs of Functions (Cambridge (CIE) A Level Maths: Pure 1): Exam Questions

Exam code: 9709

2 hours30 questions
Easy
Medium
Hard
1
3 marks

Sketch the graph of y = 6x - 12, giving the coordinates of the points where the graph meets the coordinate axes.

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2
3 marks

Sketch the graph of y = x^{2} - 1, giving the coordinates of the points where the graph meets the coordinate axes.

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3
3 marks

Sketch the graph of y = \dfrac{1}{x}, stating the coordinates of any points where the graph meets the coordinate axes and the equations of any asymptotes.

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4
3 marks

(i) On the axes below, sketch the graphs of y = x and y = -x + 2.

Blank coordinate grid with x- and y-axes from −8 to 8, with gridlines at each integer.

(ii) Use your graphs to solve the simultaneous equations y = x and y = -x + 2.

How did you do?

5a
2 marks

The function \text{f} is defined by

\text{f}(x) = x^{2} + 3x - 4

for x \in \mathbb{R}.

(i) Write down the value of \text{f}(0).

(ii) Factorise \text{f}(x).

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5b
3 marks

Hence sketch the graph of y = \text{f}(x), giving the coordinates of the points where the graph meets the coordinate axes.

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6
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3 marks

y is proportional to x. When x = 2, y = 10.

(i) Find the constant of proportionality.

(ii) Sketch the graph of y against x.

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7
3 marks

By sketching the graphs of y = x^{3} and y = \dfrac{1}{x} on the same diagram, show that the equation x^{3} = \dfrac{1}{x} has exactly two real solutions.

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8
3 marks

The diagram shows the circle with equation (x - 3)^{2} + (y + 2)^{2} = 9.

On the diagram, draw three straight lines through the point (6, -4) to show that a line through this point can meet the circle at no points, at exactly one point, or at two points.

Coordinate grid from −8 to 8 showing the circle (x − 3)² + (y + 2)² = 9, centred at (3, −2) with radius 3, and the point (6, −4) marked outside the circle.

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1a
2 marks

Express 2x^{3} + 2x^{2} - 12x in the form ax(x + b)(x + c), where a, b and c are integers.

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1b
3 marks

Hence sketch the graph of y = 2x^{3} + 2x^{2} - 12x, giving the coordinates of the points where the graph meets the coordinate axes.

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2
3 marks

Sketch the graph of y = (x + 3)^{3}, giving the coordinates of the points where the graph meets the coordinate axes.

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3
2 marks

Sketch the graph of y = \dfrac{1}{x^{2}}, and write down the equations of any asymptotes.

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4a
2 marks

On the axes below, sketch the graphs of y = 3x - 2 and y = x + 4.

Blank coordinate grid with x- and y-axes from −8 to 8, with gridlines at each integer.

How did you do?

4b
1 mark

Using your graphs, or otherwise, solve the simultaneous equations 3x - y = 2 and x - y = -4.

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5
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3 marks

y is inversely proportional to x. When x = 3, y = 12.

Find the constant of proportionality and sketch the graph of y against x.

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6
3 marks

Sketch the graph of y = 2x^{2}(x + 3), giving the coordinates of the points where the graph meets the coordinate axes.

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7a
3 marks

On the same diagram, sketch the graphs of y = x(x + 2)(x - 1) and y = \dfrac{1}{x}.

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7b
1 mark

Use your diagram to determine the number of solutions of the equation x(x + 2)(x - 1) = \dfrac{1}{x}.

How did you do?

8a
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2 marks

A machine completes a calculation in a time, t seconds, that is proportional to the number of processes, p, involved. For a calculation involving 10 processes the machine takes 0.01 seconds.

Show that the constant of proportionality is 0.001.

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8b
1 mark

Hence write down an equation linking p and t.

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8c
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2 marks

Find the time it takes for the machine to complete a calculation involving 200 processes.

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8d
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2 marks

How many processes are involved in a calculation taking 2.3 seconds?

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9a
2 marks

The diagram shows the graph of y = \dfrac{a}{x}, where a > 0.

Graph of y = a/x with a > 0: two branches in the first and third quadrants, approaching the x-axis and y-axis as asymptotes.

Sketch the graph of y = \dfrac{a}{x}, where a < 0.

How did you do?

9b
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4 marks

State which graph each of the following points must lie on, and find the value of a in each case.

(i) (-4, -5)

(ii) (-0.02, 250)

How did you do?

10a
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3 marks

Solve the equation x^{3} - x^{2} - 2x + 4 = 4x + 4.

How did you do?

10b
1 mark

Write down the x-coordinates of the points of intersection of the graphs of y = x^{3} - x^{2} - 2x + 4 and y = 4x + 4.

How did you do?

10c
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2 marks

Find the y-coordinates of these points of intersection.

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10d
4 marks

On the same diagram, sketch the graphs of y = x^{3} - x^{2} - 2x + 4 and y = 4x + 4.

How did you do?

11
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4 marks

y is inversely proportional to x. When x = 2, y = 10.

Find the constant of proportionality and sketch the graph of y against x.

How did you do?

1
2 marks

Sketch the graph of y = \dfrac{-2}{x^{2}}, and write down the equations of any asymptotes.

How did you do?

2a
3 marks

On the axes below, sketch the graphs of y = x^{2} + 2x - 3 and y = x - 1.

Blank coordinate grid with x- and y-axes from −8 to 8, with gridlines at each integer.

How did you do?

2b
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2 marks

Using your graphs, or otherwise, solve the simultaneous equations y = (x + 3)(x - 1) and x - y = 1.

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3
3 marks

Sketch the graph of y = 3x^{3} - 2x^{2} - x, giving the coordinates of the points where the graph meets the coordinate axes.

How did you do?

4a
3 marks

On the same diagram, sketch the graphs of y = x^{3} - 2x^{2} - 8x and y = \dfrac{1}{x}.

How did you do?

4b
1 mark

Use your diagram to determine the number of solutions of the equation x^{3} - 2x^{2} - 8x = \dfrac{1}{x}.

How did you do?

5a
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2 marks

A machine completes a calculation in a time, t seconds, that is proportional to the square of the number of processes, p, involved. For a calculation involving 8 processes the machine takes 0.032 seconds.

Find an equation linking p and t.

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5b
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2 marks

How many processes are involved in a calculation taking 0.2 seconds?

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5c
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2 marks

Find the time it takes for the machine to complete a calculation involving 30 processes.

How did you do?

6a
2 marks

The diagram shows the graph of y = \dfrac{a}{x^{2}}, where a > 0.

Graph of y = a/x² with a > 0: two branches above the x-axis in the first and second quadrants, symmetric about the y-axis, approaching both axes as asymptotes.

Sketch the graph of y = \dfrac{a}{x^{2}}, where a < 0.

How did you do?

6b
2 marks

Given that m is a negative real number, state, with a reason, which graph passes through the point (m, m^{4}).

How did you do?

7a
3 marks

On the same diagram, sketch the graphs of y = \dfrac{1}{x^{2}} and y = \dfrac{-3}{x^{2}}.

How did you do?

7b
2 marks

Write down the equation(s) of any lines of symmetry and asymptotes for the two graphs in part (a).

How did you do?

8a
3 marks

On the axes below, sketch the graphs of y = (x - 1)^{2} and y = 2 - x^{2} - x.

Blank coordinate grid with x- and y-axes from −8 to 8, with gridlines at each integer.

How did you do?

8b
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2 marks

Using your graphs, or otherwise, find the solutions of the equation x^{2} - 2x + 1 = 2 - x^{2} - x.

How did you do?

9
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4 marks

y is inversely proportional to the square of x. When x = 4, y = 8.

Find the constant of proportionality and sketch the graph of y against x.

How did you do?

10a
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4 marks

A machine completes a calculation in a time, t seconds, that is proportional to the cube root of the number of processes, p, involved. For a calculation involving 8 processes the machine takes 6.4 \times 10^{-4} seconds.

How many processes are involved in a calculation taking 1.28 \times 10^{-3} seconds?

How did you do?

10b
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2 marks

Find the time it takes for the machine to complete a calculation involving 250 processes.

How did you do?

11a
3 marks

On separate diagrams, sketch the graphs of y = \dfrac{a}{x^{2}}, where a > 0, and y = \dfrac{a}{x^{2}}, where a < 0.

How did you do?

11b
3 marks

One of the graphs passes through the point with coordinates (m, m^{6}).

Write a in terms of m and, justifying your answer, state which graph this point lies on.

How did you do?