(a)Complete the table of values for y = x3 - 9x.(3)
(b)Draw the graph of y = x3 - 9x for -3 ≤ x ≤ 3 on the grid provided.(2)
(c)By drawing the line y = 5 on the same grid, use your graph to find estimates for the three solutions of x3 - 9x = 5.(3)
(d)Write down the three values of x for which x3 - 9x = 0.(1)
(Total for Question 9 is 9 marks)
10
The time, t hours, taken to fill a water tank is inversely proportional to the number of pumps, n, used. When n = 4, t = 3.
(a)Find a formula for t in terms of n.(2)
(b)Complete the table of values for t = 12/n. n : 1 , 2 , 3 , 4 , 6 t : _ , _ , _ , 3 , _(4)
(c)State one feature of the graph of t against n, for n > 0.(1)
(Total for Question 10 is 7 marks)
11
The graph of y = x3 - 7x passes through the point (2, -6).
(a)Show that (2, -6) lies on the curve.(1)
(b)Use the symmetry of the graph to write down another point on the curve.(1)
(Total for Question 11 is 2 marks)
12
The graph of y = 6/x is a reciprocal curve.
(a)Write down the equations of the two lines of symmetry of the graph.(2)
(b)State the equation of the vertical line and the equation of the horizontal line that the graph never crosses.(2)
(Total for Question 12 is 4 marks)
13
By first finding where the graph crosses the axes, sketch the graph of y = (x + 4)(x - 1)(x - 3), showing the coordinates of every point where the curve crosses the x-axis and the y-axis.
(Total for Question 13 is 5 marks)
14
The graphs of y = x3 and y = 4/x intersect at one point where x > 0.
(a)Show that the x-coordinate of this point satisfies x4 = 4.(2)
(b)Hence find the exact value of x at this point.(2)
(Total for Question 14 is 4 marks)
15
The curve C has equation y = 5/x - 2.
(a)Write down the equations of the two asymptotes of C.(2)
(b)Find the coordinates of the point where C crosses the x-axis.(2)
(c)Given that the point (1, p) lies on C, find the value of p.(1)
(Total for Question 15 is 5 marks)
Mark scheme · 5.10D Drawing Cubic and Reciprocal Graphs: Fluency and Exam Drill
Question 1
B1 y = -2 at x = -1
B1 y = -1 at x = 0
B1 y = 0 at x = 1
Answer: x = -1: y = -2; x = 0: y = -1; x = 1: y = 0
Question 2
B1 all 5 points from the table plotted correctly, ft candidate's table from Q1
B1 single smooth curve drawn through all the points, correct increasing s-shape, not a series of straight lines
Answer: Smooth increasing curve through (-2,-9), (-1,-2), (0,-1), (1,0), (2,7)
Question 3
B1 y = -12 at x = -2
B1 y = -4 at x = 0
B1 y = 4 at x = 2
Answer: x = -2: y = -12; x = 0: y = -4; x = 2: y = 4
Question 4
(a) B1 all 5 points from the table plotted correctly, ft candidate's table from Q3
(a) B1 single smooth curve drawn through all the points, correct increasing s-shape with no turning points
(a) Answer: Smooth curve through (-2,-12), (-1,-5), (0,-4), (1,-3), (2,4)
(a) B1 branch in the region x<0 plotted correctly through (-4,-1), (-2,-2), (-1,-4), ft candidate's table
(a) B1 branch in the region x>0 plotted correctly through (1,4), (2,2), (4,1), ft candidate's table
(a) B1 two separate smooth curved branches drawn, correctly getting closer to both axes without touching or crossing them, and not joined across x=0
(a) Answer: Two branches: one through (-4,-1), (-2,-2), (-1,-4); one through (1,4), (2,2), (4,1)
(b) B1 correct explanation that division by zero is undefined, e.g. 4/0 has no value, oe
(b) Answer: x cannot equal 0 because 4/0 is undefined (division by zero has no value)
Question 7
(a) B1 A cao
(a) Answer: A
(b) B1 B cao
(b) Answer: B
(c) B1 C cao
(c) Answer: C
Question 8
(a) M1 takes out a factor of x to give x(x2 - 9)
(a) A1 correctly factorises the difference of two squares to give x(x-3)(x+3), cso
(a) Answer: y = x(x - 3)(x + 3)
(b) B1 (-3, 0)
(b) B1 (0, 0)
(b) B1 (3, 0)
(b) Answer: (-3, 0), (0, 0), (3, 0)
(c) B1 (0, 0) cao
(c) Answer: (0, 0)
Question 9
(a) B1 y = 10 at x = -2
(a) B1 y = 0 at x = 0
(a) B1 y = -10 at x = 2
(a) Answer: x=-2: y=10; x=0: y=0; x=2: y=-10
(b) B1 all 7 points plotted correctly, ft candidate's table
(b) B1 single smooth curve with two turning points, crossing the x-axis at (-3,0), (0,0) and (3,0)
(b) Answer: Smooth curve through (-3,0), (-2,10), (-1,8), (0,0), (1,-8), (2,-10), (3,0) with a local maximum near x=-1.7 and a local minimum near x=1.7
(c) B1 x = awrt -2.7 (accept -2.9 to -2.5), ft consistent with candidate's own graph
(c) B1 x = awrt -0.6 (accept -0.8 to -0.4), ft consistent with candidate's own graph
(c) B1 x = awrt 3.2 (accept 3.1 to 3.4), ft consistent with candidate's own graph
(c) Answer: x = -2.7, x = -0.6, x = 3.2 (each awrt 1 dp)
(d) B1 x = -3, 0, 3 (all three), ft from graph or algebra, cao
(d) Answer: x = -3, 0, 3
Question 10
(a) M1 sets up t = k/n and substitutes n=4, t=3 to find k=12
(c) B1 any correct feature, e.g. as n increases t decreases; or the curve gets closer to the n-axis but never touches it; or the graph only exists for n > 0 in this context
(c) Answer: As the number of pumps increases, the time taken decreases (the curve approaches but never touches the n-axis)
Question 11
(a) B1 substitutes x=2 to get y = 8-14 = -6, confirming the point lies on the curve
(a) Answer: 23 - 7(2) = 8 - 14 = -6, so (2,-6) lies on the curve
(b) B1 (-2, 6), by the rotational symmetry of a cubic of this form about the origin, oe
(b) Answer: (-2, 6)
Question 12
(a) B1 y = x
(a) B1 y = -x
(a) Answer: y = x and y = -x
(b) B1 x = 0
(b) B1 y = 0
(b) Answer: x = 0 and y = 0
Question 13
B1 x-intercept (-4, 0)
B1 x-intercept (1, 0)
B1 x-intercept (3, 0)
B1 y-intercept (0, 12)
B1 correct overall cubic shape with two turning points: curve rises from bottom-left, crosses upward at x=-4, turns to a local maximum, falls through x=1, turns to a local minimum, then rises through x=3 to top-right
Answer: x-intercepts (-4,0), (1,0), (3,0); y-intercept (0,12); positive cubic with two turning points
Question 14
(a) M1 sets x3 = 4/x and multiplies both sides by x
(a) A1 correctly obtains x4 = 4, cso
(a) Answer: x4 = 4
(b) M1 takes the square root of both sides to get x2 = 2 (positive root, since x2 ≥ 0)
(b) A1 x = √2 oe, cao (positive root only, since x > 0)