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Grade 5 · Foundation and Higher · free GCSE maths worksheet

Graphs of cubic and reciprocal functions

Recognising and sketching their shapes.

15 original questions, 46 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

Complete the table of values for y = x3 - 2

x-2-1012
y-10-26
2 marks
Show answer
x = -1) -3 x = 1) -1
2

Here are four equations.

A  y = 3x + 2     B  y = x2     C  y = 4x     D  y = x3

  1. Write down the letter of the equation whose graph is a reciprocal curve.
  2. Write down the letter of the equation whose graph is a straight line.
2 marks
Show answer
a) C b) A
3

Here are four equations.

  A   y = 2x + 1     B   y = 5x     C   y = x3     D   y = x2 − 4

Write down the letter of the equation whose graph

  1. is a curve with two separate branches,
  2. is an S-shaped curve that passes through the origin,
  3. is a straight line.
3 marks
Show answer
a) B b) C c) A
4

A graph has equation y = x3

  1. Work out the value of y when x = −3.
  2. Does the point (−2, 8) lie on the graph? You must show how you get your answer.
  3. Work out the value of x when y = 64.
3 marks
Show answer
a) -27 b) No, y = -8 when x = -2 c) 4
5

The diagram shows four sketch graphs, labelled A, B, C and D.

  1. Write down the letter of the graph of y = x3
  2. Write down the letter of the graph of y = 2x
xyAxyBxyCxyD
2 marks
Show answer
a) C b) D
6

Complete the table of values for y = 12x

x1234612
y1242
2 marks
Show answer
x = 2) 6 x = 4) 3 x = 12) 1
7

The graph of y = x3 is drawn for all values of x.

  1. Write down the coordinates of the point where the graph crosses the x-axis.
  2. Is y positive or negative when x is a large negative number such as −100? Give a reason.
  3. Describe the shape of the graph as x goes from large negative values to large positive values.
3 marks
Show answer
a) (0, 0) b) negative; (-100)3 is negative c) rising S-shaped curve, bottom left to top right, passing through the origin
8

The diagram shows five sketch graphs, labelled A, B, C, D and E.

Match each equation below to the letter of its graph.

  1. y = -x2
  2. y = x3
  3. y = 5x
xyAxyBxyCxyDxyE
3 marks
Show answer
a) C b) D c) E
9

The curve with equation y = kx passes through the point (4, 3).

  1. Work out the value of k.
  2. Work out the value of y when x = −6.
  3. Does the point (−3, 4) lie on the curve? You must show how you get your answer.
4 marks
Show answer
a) 12 b) -2 c) No, when x = -3, y = -4
10

A curve has equation y = 8x

  1. Explain why there is no point on the curve where x = 0
  2. Work out the value of y when x = -4
  3. Does the point (2, 4) lie on the curve? You must show how you get your answer.
3 marks
Show answer
a) You cannot divide 8 by 0, so there is no value of y when x = 0 b) -2 c) Yes, because 8/2 = 4
11
  1. Complete the table of values for y = x3 - 3x
    x-2-1012
    y-2-2
  2. Explain how the table shows that the graph of y = x3 - 3x passes through the origin.
3 marks
Show answer
a (x = -1)) 2 a (x = 0)) 0 a (x = 2)) 2 b) When x = 0, y = 0, so the curve passes through (0, 0)
12

A curve has equation y = 10x

  1. Work out the value of y when x = 0.5
  2. Work out the value of y when x = −5
  3. Explain what happens to y as x gets very large, and why this means the curve never touches the x-axis.
4 marks
Show answer
a) 20 b) -2 c) y gets closer to 0 but never reaches it
13
  1. Complete the table of values for y = x3 + 2
    x-2-1012
    y-6210
  2. On the grid, draw the graph of y = x3 + 2 for values of x from -2 to 2
−2−112−8−6−4−224681012Oxy
4 marks
Show answer
a (x = -1)) 1 a (x = 1)) 3 b) correct smooth cubic curve through (-2, -6), (-1, 1), (0, 2), (1, 3), (2, 10)
14
  1. Complete the table of values for y = x3 - 2x2
    x-10123
    y09
  2. Use the table to write down two values of x for which y = 0
4 marks
Show answer
a (x = -1)) -3 a (x = 1)) -1 a (x = 2)) 0 b) x = 0 and x = 2
15
  1. On the axes, sketch the graph of y = 6x for negative and positive values of x
  2. Write down the coordinates of any points where the graph crosses the coordinate axes.
−6−5−4−3−2−1123456−6−5−4−3−2−1123456Oxy
4 marks
Show answer
a) reciprocal curve with one branch in the first quadrant and one branch in the third quadrant, approaching but never touching the axes b) There are no such points - the graph never crosses the axes

Every question here is original and written by Robinson Tuition. The grade follows the Edexcel grade descriptors and suits AQA and OCR too; this is not an exam-board paper.

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