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GCSE · Algebra - Graphs

5.10 Drawing Other Graphs: Cubic/Reciprocal

EDEXCEL 1MA1 · Calculator allowed · about 100 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

The graph of y = x³ is to be drawn for -2 ≤ x ≤ 2.

(a)Complete the table of values for y = x^3.
x : -2 , -1 , 0 , 1 , 2
y : -8 , _ , 0 , _ , 8(2)

(b)On the grid, plot the points from your table and join them with a smooth curve.(2)

(c)Write down the coordinates of the point where the curve crosses the y-axis.(1)

(Total for Question 1 is 5 marks)

For each equation below, state whether its graph is linear, quadratic, cubic or reciprocal.

(a)y = 4x - 3(1)

A) Linear
B) Quadratic
C) Cubic
D) Reciprocal

(b)y = x² + 5(1)

A) Linear
B) Quadratic
C) Cubic
D) Reciprocal

(c)y = x³ - 6x(1)

A) Linear
B) Quadratic
C) Cubic
D) Reciprocal

(d)y = 10/x(1)

A) Linear
B) Quadratic
C) Cubic
D) Reciprocal

(Total for Question 2 is 4 marks)

The graph of y = 12/x is to be drawn for -6 ≤ x ≤ 6 (x is not equal to 0).

(a)Complete the table of values for y = 12/x.
x : -6 , -4 , -2 , -1 , 1 , 2 , 4 , 6
y : -2 , _ , -6 , -12 , 12 , _ , 3 , 2(2)

(b)Explain why the table has no value when x = 0.(1)

(c)Use your calculator to find the value of y when (i) x = 0.5 and (ii) x = -0.8.(2)

(d)State whether the point (0, 0) lies on the graph of y = 12/x, giving a reason.(1)

(Total for Question 3 is 6 marks)

Using your completed table of values for y = 12/x from Question 3, draw the graph on the grid provided.

(a)Plot the 8 points from the table for y = 12/x and draw two smooth curve branches.(2)

(b)State the order of rotational symmetry of the graph of y = 12/x.(1)

(Total for Question 4 is 3 marks)

The graph of y = x³ - 2x is to be drawn for -3 ≤ x ≤ 3.

(a)Complete the table of values for y = x³ - 2x.
x : -3 , -2 , -1 , 0 , 1 , 2 , 3
y : -21 , _ , 1 , 0 , _ , 4 , 21(2)

(b)On the grid, plot the points from your table and join them with a smooth curve.(2)

(Total for Question 5 is 4 marks)

Use your graph of y = x³ - 2x from Question 5 to answer these questions.

(a)Use the graph to find estimates for the value of y when (i) x = 2.5 and (ii) x = -1.5.(2)

(b)Use the graph to find the three solutions of x³ - 2x = 0.(2)

(c)Write down the coordinates of the point where the curve crosses the y-axis.(1)

(Total for Question 6 is 5 marks)

The graph of y = x³ - 2x is drawn on the same grid as Question 5.

(a)By drawing the line y = 4 on the grid, use the graphs to find the solution of x³ - 2x = 4.(2)

(b)Explain how you can tell from the grid that this is the only solution to x³ - 2x = 4 in the range -3 ≤ x ≤ 3.(1)

(Total for Question 7 is 3 marks)

A minibus costs a total of GBP 180 to hire. The cost per person, C pounds, when the cost is shared equally between n people is given by C = 180/n.

(a)Complete the table of values for C = 180/n.
n : 1 , 2 , 3 , 4 , 5 , 6 , 9 , 10
C : 180 , _ , 60 , _ , 36 , _ , 20 , _(3)

(b)On the grid, plot the points from the table and draw a smooth curve for C against n.(2)

(c)Use the graph, or otherwise, to estimate the cost per person if 8 people share the minibus.(1)

(d)Find the least number of people needed to share the minibus so that the cost per person is below GBP 15.(2)

(Total for Question 8 is 8 marks)

For each statement, state whether it is True or False, giving a brief reason.

(a)The graph of y = 1/x has a value when x = 0.(1)

(b)A cubic graph can cross the x-axis at most three times.(1)

(c)The graph of y = x³ is symmetrical about the y-axis.(1)

(d)For a>0, the two branches of the graph y = a/x lie in opposite quadrants.(1)

(e)As x gets very large, the value of 1/x gets closer to 0.(1)

(f)Every cubic graph has a turning point (a local maximum or minimum).(1)

(Total for Question 9 is 6 marks)

A cubic graph has the equation y = x³ + 2.

(a)Write down the coordinates of the y-intercept of the graph.(1)

(b)State whether the value of y increases throughout as x increases, giving a reason.(1)

(c)Describe the single transformation that maps the graph of y = x³ onto the graph of y = x³ + 2.(1)

(Total for Question 10 is 3 marks)

Match each equation to the description of its graph. Equations: y = 2x + 5, y = 8 - x², y = x³ + x, y = -6/x.

(a)y = 2x + 5(1)

i) A straight line with a positive gradient
ii) An upside-down U-shaped curve with one maximum point
iii) A wave-shaped curve through the origin that increases throughout
iv) A two-branch curve lying in the second and fourth quadrants

(b)y = 8 - x²(1)

i) A straight line with a positive gradient
ii) An upside-down U-shaped curve with one maximum point
iii) A wave-shaped curve through the origin that increases throughout
iv) A two-branch curve lying in the second and fourth quadrants

(c)y = x³ + x(1)

i) A straight line with a positive gradient
ii) An upside-down U-shaped curve with one maximum point
iii) A wave-shaped curve through the origin that increases throughout
iv) A two-branch curve lying in the second and fourth quadrants

(d)y = -6/x(1)

i) A straight line with a positive gradient
ii) An upside-down U-shaped curve with one maximum point
iii) A wave-shaped curve through the origin that increases throughout
iv) A two-branch curve lying in the second and fourth quadrants

(e)Explain why the graph of y = x³ + x is increasing for all values of x.(1)

(Total for Question 11 is 5 marks)

A cubic graph has the equation y = (x - 1)(x + 3)(x - 2).

(a)Find the three x-coordinates of the points where the graph crosses the x-axis.(2)

(b)Find the coordinates of the point where the graph crosses the y-axis.(2)

(Total for Question 12 is 4 marks)

A car travels a fixed distance of 240 km. The time taken, T hours, at an average speed of S km/h is given by T = 240/S.

(a)Complete the table of values for T = 240/S.
S : 20 , 30 , 40 , 48 , 60 , 80 , 100 , 120
T : 12 , _ , 6 , _ , 4 , _ , 2.4 , _(3)

(b)On the grid, plot the points from the table and draw a smooth curve for T against S.(2)

(c)Use the graph to estimate the speed needed to complete the journey in 3 hours.(1)

(d)Calculate the exact time taken, in hours and minutes, at an average speed of 100 km/h.(2)

(Total for Question 13 is 8 marks)

A reciprocal graph has the equation y = 2/x + 3.

(a)State the equation of the horizontal asymptote of the graph.(1)

(b)State the equation of the vertical asymptote of the graph.(1)

(c)Find the exact coordinates of the point where the graph crosses the x-axis.(2)

(Total for Question 14 is 4 marks)

The pressure, P pascals, of a fixed mass of gas is inversely proportional to its volume, V m³, so that P = k/V for a constant k. When V = 2, P = 150.

(a)Find the value of k.(2)

(b)Find the value of P when V = 5.(2)

(c)Describe the shape of the graph of P against V for V > 0.(1)

(d)State what happens to the pressure P if the volume V is doubled.(1)

(Total for Question 15 is 6 marks)

The graphs of y = x³ - 4x and y = x intersect at three points.

(a)Show that the x-coordinates of the points of intersection satisfy x³ - 5x = 0.(1)

(b)Use the graphs, or otherwise, to find the three solutions of x³ - 5x = 0.(2)

(c)Verify that x = 0 is a solution by substituting it into both y = x³ - 4x and y = x.(1)

(d)Write down the y-coordinate of the intersection point where x is positive, giving your answer to 1 decimal place.(1)

(Total for Question 16 is 5 marks)

Show that the equation x³ - 3x - 6 = 0 has a solution between x = 2 and x = 3.

(Total for Question 17 is 3 marks)

The graph of y = x³ is reflected to give the graph of y = -x^3.

(a)Describe the single transformation that maps the graph of y = x³ onto the graph of y = -x^3.(1)

(b)Write down the coordinates of the y-intercept of the graph of y = -x^3.(1)

(c)State whether y = -x³ is increasing or decreasing as x increases, giving a reason.(1)

(Total for Question 18 is 3 marks)

The graph of y = x³ - kx passes through the point (3, 18).

(a)Find the value of k.(2)

(b)Using this value of k, find the value of y when x = -3.(2)

(c)Using this value of k, find the value of y when x = 5.(2)

(Total for Question 19 is 6 marks)

The table shows values of f(x) = x³ - 3x + 1 for integer values of x from -2 to 2.
x : -2 , -1 , 0 , 1 , 2
f(x) : -1 , 3 , 1 , -1 , 3

(a)Show that f(-1) = 3.(1)

(b)Using the table, write down the three intervals (each of width 1) in which the equation x³ - 3x + 1 = 0 has a solution.(3)

(c)State the total number of real solutions to the equation x³ - 3x + 1 = 0.(1)

(Total for Question 20 is 5 marks)

Mark scheme · 5.10 Drawing Other Graphs: Cubic/Reciprocal

Question 1

(a) B1 y = -1 at x = -1
(a) B1 y = 1 at x = 1
(a) Answer: x=-1: -1, x=1: 1
(b) B1 all 5 points plotted correctly ft from table
(b) B1 smooth s-shaped curve drawn through all points, not a series of straight lines
(b) Answer: Smooth increasing s-shaped curve through the 5 points
(c) B1 (0, 0) cao
(c) Answer: (0, 0)

Question 2

(a) B1 A (Linear)
(a) Answer: A) Linear
(b) B1 B (Quadratic)
(b) Answer: B) Quadratic
(c) B1 C (Cubic)
(c) Answer: C) Cubic
(d) B1 D (Reciprocal)
(d) Answer: D) Reciprocal

Question 3

(a) B1 y = -3 at x = -4
(a) B1 y = 6 at x = 2
(a) Answer: x=-4: -3, x=2: 6
(b) B1 12/0 is undefined (division by zero is not possible) oe
(b) Answer: Because dividing by 0 is undefined, so y has no value when x = 0
(c) B1 y = 24 at x = 0.5
(c) B1 y = -15 at x = -0.8
(c) Answer: y = 24 and y = -15
(d) B1 No, because x = 0 is not defined for this graph oe
(d) Answer: No, because the graph is undefined at x = 0

Question 4

(a) B1 all 8 points plotted correctly ft from Question 3's table
(a) B1 two separate smooth curve branches drawn, one in the top-right region and one in the bottom-left region, not joined together
(a) Answer: Two smooth curve branches, one through the positive x-values and one through the negative x-values, neither touching the axes
(b) B1 order 2
(b) Answer: Order 2

Question 5

(a) B1 y = -4 at x = -2
(a) B1 y = -1 at x = 1
(a) Answer: x=-2: -4, x=1: -1
(b) B1 all 7 points plotted correctly ft from table
(b) B1 smooth wave-shaped curve drawn through all points, with a local maximum and a local minimum
(b) Answer: Smooth wave-shaped cubic curve through the 7 points, with a local maximum near x=-0.8 and a local minimum near x=0.8

Question 6

(a) B1 y = awrt 10.6 (range 10.3 to 10.9), ft from candidate's graph
(a) B1 y = awrt -0.4 (range -0.6 to -0.2), ft from candidate's graph
(a) Answer: y = awrt 10.6 and y = awrt -0.4
(b) B1 x = 0 cao
(b) B1 x = awrt 1.4 and x = awrt -1.4 (range 1.3 to 1.5 and -1.5 to -1.3), both required, ft
(b) Answer: x = 0, x = awrt 1.4, x = awrt -1.4
(c) B1 (0, 0) cao
(c) Answer: (0, 0)

Question 7

(a) M1 line y = 4 drawn on the grid, correctly identifying the intersection with the cubic curve
(a) A1 x = 2 cao
(a) Answer: x = 2
(b) B1 the line y = 4 only crosses the cubic curve once within the given range oe
(b) Answer: The line y = 4 crosses the curve at only one point within -3 ≤ x ≤ 3

Question 8

(a) M1 at least one correct calculation shown, e.g. 180 / 4 = 45
(a) A1 at least two of the four missing values correct
(a) A1 all four missing values correct: 90, 45, 30, 18
(a) Answer: n=2: 90, n=4: 45, n=6: 30, n=10: 18
(b) B1 all 8 points plotted correctly ft from table
(b) B1 smooth decreasing curve drawn through all points, getting closer to the n-axis but never touching it
(b) Answer: Smooth decreasing reciprocal-shaped curve through the 8 points
(c) B1 awrt GBP 22.50 (range 22 to 23), ft from candidate's graph
(c) Answer: GBP 22.50
(d) M1 sets up and tests 180/n < 15, e.g. checking n=12 gives exactly 15 and n=13 gives below 15
(d) A1 n = 13 cao
(d) Answer: 13 people

Question 9

(a) B1 False, because division by zero is undefined oe
(a) Answer: False
(b) B1 True, because a cubic equation has at most three real roots oe
(b) Answer: True
(c) B1 False, it has rotational symmetry about the origin, not reflective symmetry in the y-axis oe
(c) Answer: False
(d) B1 True, one branch lies in the first quadrant and the other in the third quadrant oe
(d) Answer: True
(e) B1 True oe
(e) Answer: True
(f) B1 False, e.g. y=x³ has no turning point oe
(f) Answer: False

Question 10

(a) B1 (0, 2) cao
(a) Answer: (0, 2)
(b) B1 Yes, because it has the same basic shape as y=x³, which increases throughout, just shifted up oe
(b) Answer: Yes, y increases throughout
(c) B1 translation by vector (0, 2), i.e. 2 units up, oe
(c) Answer: Translation 2 units up (vector (0,2))

Question 11

(a) B1 i
(a) Answer: i) straight line, positive gradient
(b) B1 ii
(b) Answer: ii) n-shaped curve with a maximum
(c) B1 iii
(c) Answer: iii) wave-shaped increasing cubic curve
(d) B1 iv
(d) Answer: iv) two-branch reciprocal curve in quadrants 2 and 4
(e) B1 both x³ and x individually increase as x increases, so their sum always increases oe
(e) Answer: Both x³ and x increase as x increases, so x³ + x increases throughout

Question 12

(a) M1 sets each factor equal to 0
(a) A1 x = 1, x = -3, x = 2, all correct (any order)
(a) Answer: x = 1, x = -3, x = 2
(b) M1 substitutes x = 0 into y = (x-1)(x+3)(x-2)
(b) A1 (0, 6) cao
(b) Answer: (0, 6)

Question 13

(a) M1 at least one correct calculation shown, e.g. 240 / 48 = 5
(a) A1 at least two of the four missing values correct
(a) A1 all four missing values correct: 8, 5, 3, 2
(a) Answer: S=30: 8, S=48: 5, S=80: 3, S=120: 2
(b) B1 all 8 points plotted correctly ft from table
(b) B1 smooth decreasing curve drawn through all points, approaching but never touching the S-axis
(b) Answer: Smooth decreasing reciprocal-shaped curve through the 8 points
(c) B1 awrt 80 km/h (range 76 to 84), ft from candidate's graph
(c) Answer: 80 km/h
(d) M1 240 / 100 = 2.4 (hours)
(d) A1 2 hours 24 minutes cao
(d) Answer: 2 hours 24 minutes

Question 14

(a) B1 y = 3
(a) Answer: y = 3
(b) B1 x = 0
(b) Answer: x = 0
(c) M1 sets 2/x + 3 = 0 and rearranges to 2/x = -3
(c) A1 x = -2/3 oe (e.g. -0.667), so (-2/3, 0)
(c) Answer: (-2/3, 0)

Question 15

(a) M1 substitutes P=150 and V=2 into P=k/V, e.g. 150=k/2
(a) A1 k = 300 cao
(a) Answer: k = 300
(b) M1 substitutes k=300 and V=5 into P=k/V
(b) A1 P = 60 cao (units: pascals)
(b) Answer: P = 60 pascals
(c) B1 a curve that continually decreases as V increases, getting closer to but never touching either axis oe
(c) Answer: A decreasing reciprocal-shaped curve that never touches the axes
(d) B1 P is halved oe
(d) Answer: P is halved

Question 16

(a) B1 x³-4x=x => x³-4x-x=0 => x³-5x=0 cso
(a) Answer: x³ - 5x = 0 (shown)
(b) B1 x = 0 cao
(b) B1 x = awrt 2.2 and x = awrt -2.2 (range 2.1 to 2.3 and -2.3 to -2.1), both required, ft
(b) Answer: x = 0, x = awrt 2.2, x = awrt -2.2
(c) B1 y=0³-4(0)=0 and y=0, both equal 0 cao
(c) Answer: Both give y = 0, confirming x = 0 is a solution
(d) B1 y = awrt 2.2, ft from part (b)
(d) Answer: y = 2.2

Question 17

M1 substitutes x=2 into x³-3x-6, obtaining -4
M1 substitutes x=3 into x³-3x-6, obtaining 12
dM1 correct conclusion referencing the change of sign (from negative to positive), so a root lies between x=2 and x=3, cso
Answer: f(2) = -4 and f(3) = 12; change of sign confirms a solution lies between x=2 and x=3 (shown)

Question 18

(a) B1 reflection in the x-axis oe
(a) Answer: Reflection in the x-axis
(b) B1 (0, 0) cao
(b) Answer: (0, 0)
(c) B1 decreasing throughout, e.g. because y=x³ is increasing and it has been reflected in the x-axis oe
(c) Answer: Decreasing throughout

Question 19

(a) M1 substitutes x=3, y=18 to form 3³-3k=18 oe
(a) A1 k = 3 cao
(a) Answer: k = 3
(b) M1 substitutes x=-3 and k=3 into y=x³-3x
(b) A1 y = -18 cao
(b) Answer: y = -18
(c) M1 substitutes x=5 and k=3 into y=x³-3x
(c) A1 y = 110 cao
(c) Answer: y = 110

Question 20

(a) B1 (-1)³-3(-1)+1=-1+3+1=3 cso
(a) Answer: 3 (shown)
(b) B1 between x=-2 and x=-1 (sign changes from -1 to 3)
(b) B1 between x=0 and x=1 (sign changes from 1 to -1)
(b) B1 between x=1 and x=2 (sign changes from -1 to 3)
(b) Answer: Between x=-2 and x=-1; between x=0 and x=1; between x=1 and x=2
(c) B1 3, ft from part (b)
(c) Answer: 3 real solutions

Drawing Other Graphs: Cubic/Reciprocal - Worksheets, Questions and Revision