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5.10H Drawing Other Graphs: Cubic/Reciprocal: Higher Tier Practice

EDEXCEL Edexcel 1MA1 · Calculator allowed · about 75 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Sketch the graph of y = x³ for -2 ≤ x ≤ 2. Mark clearly the points where x = -1, 0 and 1 and give their coordinates.

(a)Mark the points for x = -1, 0 and 1 and write their coordinates.(2)

(Total for Question 1 is 2 marks)

The curve C has equation y = (x - 2)³ + 1.
State the translation that maps y = x³ to the curve C. Hence find the coordinates of the point on C that corresponds to x = 2 on y = x³.

(a)State the translation.(1)

(b)Find the coordinates of the point on C that corresponds to the point (2, 8) on y = x³.(2)

(Total for Question 2 is 3 marks)

Sketch the reciprocal curve y = 2/x for -6 ≤ x ≤ 6, excluding x = 0. On the axes, indicate clearly the vertical and horizontal asymptotes and state their equations.

(a)Indicate the vertical and horizontal asymptotes and give their equations.(2)

(b)State the coordinates of the point where the curve crosses the line y = 1.(2)

(Total for Question 3 is 4 marks)

The graphs of y = x³ - 3x and y = 2x are drawn on the same axes.
(a) Show that the x-coordinates of the intersection points satisfy x³ - 5x = 0. (b) Hence find all intersection points.

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

(b)Hence find all intersection points (give coordinates).(3)

(Total for Question 4 is 4 marks)

A transformed reciprocal curve has equation y = -3/(x + 1) + 4.
(a) State the vertical and horizontal asymptotes. (b) Find the x-intercept of this curve.

(a)State the vertical and horizontal asymptotes.(2)

(b)Find the x-intercept (give as an exact value).(2)

(Total for Question 5 is 4 marks)

Show that the cubic y = x³ - 6x² + 11x - 6 factorises as (x - 1)(x - 2)(x - 3). Hence, sketch the cubic and state the x-intercepts.

(a)Show that the cubic factorises as stated.(2)

(b)Hence sketch the cubic and state the x-intercepts.(2)

(Total for Question 6 is 4 marks)

Consider the family of cubics y = x³ + kx² where k is a real constant.
(a) Show that x = 0 is always a root. (b) For k = -3 find the other two roots and classify the nature of the turning points of the cubic (local max, local min or point of inflection).

(a)Show that x = 0 is always a root.(1)

(b)For k = -3 find the other two roots. Then, by working out y at x = -1, 0, 1, 2, 3 and 4 and comparing neighbouring values, say whether the graph has a local maximum or a local minimum at x = 0 and at x = 2.(4)

(Total for Question 7 is 5 marks)

Show that the curve y = 1/(x²) cannot have a turning point for x > 0. Then explain briefly what this implies about the shape of the graph for x > 0.

(a)Show that the curve cannot have a turning point for x > 0.(2)

(b)Explain briefly what this implies about the shape of the graph for x > 0.(2)

(Total for Question 8 is 4 marks)

Mark scheme · 5.10H Drawing Other Graphs: Cubic/Reciprocal: Higher Tier Practice

Question 1

(a) B1 (-1, -1), (0, 0) and (1, 1) all correctly stated
(a) B1 sketch shows cubic shape (increasing, point of inflection at origin) with approximate shape through marked points
(a) Answer: (-1, -1), (0, 0), (1, 1)

Question 2

(a) B1 translation 2 units right and 1 unit up oe
(a) Answer: Translate 2 units to the right and 1 unit up.
(b) M1 apply translation to (2, 8): (2+2, 8+1) or equivalent method
(b) A1 (4, 9) cao
(b) Answer: (4, 9)

Question 3

(a) B1 vertical asymptote x = 0 stated
(a) B1 horizontal asymptote y = 0 stated
(a) Answer: Vertical asymptote x = 0. Horizontal asymptote y = 0.
(b) M1 solve 2/x = 1 or equivalent
(b) A1 (2, 1) cao
(b) Answer: (2, 1)

Question 4

(a) B1 substitute y = 2x into x³ - 3x = y giving x³ - 3x = 2x and rearrange to x³ - 5x = 0 oe
(a) Answer: x³ - 5x = 0
(b) M1 factor x³ - 5x = x(x² - 5) and find x = 0, x = √5, x = -√5 (allow correct factorisation shown)
(b) M1 find corresponding y by y = 2x or substitution
(b) A1 (0, 0), (√5, 2sqrt(5)), (-√5, -2sqrt(5)) cao
(b) Answer: (0, 0), (√5, 2 √5), (-√5, -2 √5)

Question 5

(a) B1 vertical asymptote x = -1 stated
(a) B1 horizontal asymptote y = 4 stated
(a) Answer: Vertical asymptote x = -1. Horizontal asymptote y = 4.
(b) M1 set y = 0 and solve -3/(x+1) + 4 = 0 leading to -3/(x+1) = -4
(b) A1 x = -1/4 cao
(b) Answer: x = -1/4

Question 6

(a) M1 either show synthetic division by x=1 then x=2 or factor by grouping leading to factors (x-1),(x-2),(x-3)
(a) A1 (x - 1)(x - 2)(x - 3) cao
(a) Answer: (x - 1)(x - 2)(x - 3)
(b) M1 sketch shows cubic crossing x-axis at three points and general shape correct
(b) A1 x-intercepts (1, 0), (2, 0), (3, 0) cao
(b) Answer: (1, 0), (2, 0), (3, 0)

Question 7

(a) B1 substitute x = 0 to give y = 0 thereby showing x=0 is a root
(a) Answer: Substitute x = 0 gives y = 0, so x = 0 is a root.
(b) M1 factor y = x³ - 3x² as x²(x - 3) or equivalent to find roots
(b) M1 identify roots x = 0 (double), x = 3 or explicitly x = 0,0,3
(b) M1 work out values e.g. y(-1) = -4, y(0) = 0, y(1) = -2, y(2) = -4, y(3) = 0 and compare neighbouring values around x = 0 and x = 2
(b) A1 x = 0 is a local maximum (0 is higher than the neighbouring values -4 and -2) and x = 2 is a local minimum (-4 is lower than the neighbouring values -2 and 0) cao
(b) Answer: Other roots: x = 0 (double root) and x = 3. Using y(-1) = -4, y(0) = 0, y(1) = -2, y(2) = -4, y(3) = 0: x = 0 is a local maximum (0 is higher than the values on either side, -4 and -2) and x = 2 is a local minimum (-4 is lower than the values on either side, -2 and 0).

Question 8

(a) M1 state that for x > 0, as x increases, x² increases, so y = 1/x² decreases (support with values, e.g. y(1) = 1, y(2) = 0.25, y(3) = 1/9)
(a) A1 conclude y is strictly decreasing throughout x > 0, so it never changes from increasing to decreasing (or vice versa) and cannot have a turning point oe
(a) Answer: For x > 0, as x increases x² increases so y = 1/x² strictly decreases throughout; since the graph never stops decreasing, it cannot have a turning point for x > 0.
(b) M1 state that the graph is strictly decreasing for x > 0 (no local max/min)
(b) A1 conclude no turning points for x>0 so graph decreases from +infinity towards 0 approaching horizontal asymptote y=0 cao
(b) Answer: For x > 0 the graph is strictly decreasing throughout: it falls from +infinity as x->0+ and approaches the horizontal asymptote y = 0 as x->infinity; there are no local maxima or minima for x>0.

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