Pure: Functions, Graphs and Transformations - Worksheets, Questions and Revision

12 original exam-style questions - 5 pages of questions with a full mark scheme - free printable PDF.

Download PDF Jump to mark scheme (page 6)Read the revision guide

Next in this strand

Pure: Trigonometric Graphs

Revision Library

revisionlibrary.co.uk

A-Level · Pure Mathematics

P17 Pure: Functions, Graphs and Transformations

EDEXCEL 9MA0 · Calculator allowed · about 135 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

The curve C has equation y = (x + 3)(x - 1)(x - 2).

(a)Write down the coordinates of the three points where C crosses the x-axis.(2)

(b)Write down the coordinate where C crosses the y-axis.(1)

(c)State the behaviour of y as x -> +infinity and as x -> -infinity.(1)

(d)Hence sketch the graph of C for -4 ≤ x ≤ 3, labelling the coordinates of the points where the curve meets the coordinate axes.(3)

(Total for Question 1 is 7 marks)

The curves C1: y = 1/x and C2: y = 1/x² are both defined for x not equal to 0.

(a)State the equations of the asymptotes of C1.(2)

(b)Sketch the graph of C1 for -3 ≤ x ≤ 3, x not equal to 0, showing the shape of the curve in each of the regions x > 0 and x < 0.(3)

(c)Explain how the graph of C2: y = 1/x² differs from the graph of C1 in the regions x > 0 and x < 0.(2)

(d)Write down the equations of the asymptotes of C2, and state the number of points at which C2 crosses either coordinate axis.(2)

(Total for Question 2 is 9 marks)

A campsite hire company found that the cost, C pounds, of hiring a bell tent for a weekend is directly proportional to the number of guests, n, that the tent must sleep. A tent that sleeps 4 guests costs 60 pounds to hire.

(a)Write down an equation connecting C and n, in terms of a constant k, and find the value of k.(2)

(b)State the cost of hiring a tent that sleeps 7 guests.(1)

(c)Sketch the graph of C against n for n > 0, labelling the gradient of the line and the coordinates of one other point on the line.(3)

(d)A different company hires the same style of tent for a fee, F pounds per weekend, that is inversely proportional to the number of weekends, w, that the customer books, for w > 0. A customer who books 3 weekends is charged a fee of 160 pounds per weekend. Find an equation connecting F and w, and sketch the graph of F against w for w > 0, labelling the equations of the asymptotes and the coordinates of one point on the curve.(5)

(Total for Question 3 is 11 marks)

A model rocket is launched from rest. For the first 4 seconds of flight, the height, h metres, reached by the rocket is proportional to the square of the time, t seconds, since launch. After 2 seconds, the rocket has reached a height of 20 metres.

(a)Find an equation connecting h and t, and state the value of the constant of proportionality.(2)

(b)Find the height reached after 3.5 seconds.(2)

(c)Sketch the graph of h against t for 0 ≤ t ≤ 4, labelling the coordinates of the endpoints of the curve.(3)

(d)The intensity, I watts per square metre, of light received from a lamp is inversely proportional to the square of the distance, x metres, from the lamp. At a distance of 2 m the intensity is 45 W/m². Find an equation connecting I and x, and hence find the distance at which the intensity is 5 W/m².(4)

(e)For a relationship of the form y = kx², state the effect on y of doubling x. For a relationship of the form y = k*√x, state the effect on y of doubling x, giving your answer as an exact multiplier.(2)

(f)The time, T seconds, for one complete swing of a simple pendulum is directly proportional to the square root of its length, L metres. A pendulum of length 4 m has a swing time of 4 seconds. Find an equation connecting T and L, and find the time for one swing of a pendulum of length 9 m.(3)

(Total for Question 4 is 16 marks)

The diagram shows a sketch of the curve y = f(x).

Figure (to be drawn): Sketch of the curve y = f(x): a single smooth curve with one minimum turning point at (2, -3), marked and labelled; the curve crosses the y-axis at the labelled point (0, 5); the curve is decreasing for x < 2 and increasing for x > 2; no other points are labelled.

(a)State the coordinates of the minimum turning point of the curve y = f(x) + 4.(2)

(b)State the coordinates of the minimum turning point of the curve y = f(x - 3).(2)

(c)The point A(0, 5) lies on the curve y = f(x). State the coordinates of the image of A under the transformation that produces the curve y = f(x + 1) - 2.(2)

(d)Given further that the curve y = f(x) crosses the x-axis only at x = a and x = b, where a < b, state, in terms of a and b, the x-intercepts of the curve y = f(x - 3).(3)

(Total for Question 5 is 9 marks)

The diagram shows a sketch of the curve y = g(x).

Figure (to be drawn): Sketch of the curve y = g(x): a single smooth curve with one maximum turning point at (-1, 6), marked and labelled; the curve crosses the x-axis at the two labelled points (-3, 0) and (2, 0); the curve is increasing for x < -1 and decreasing for x > -1; no other points are labelled.

(a)State the coordinates of the maximum turning point of the curve y = 3g(x).(2)

(b)State the coordinates of the maximum turning point of the curve y = g(2x).(2)

(c)State the coordinates of the two points where the curve y = g(x/2) crosses the x-axis.(3)

(d)Given that the maximum value of g(x) is 6, state the maximum value of the function h(x) = 2g(x) - 5, and the value of x at which this maximum occurs.(2)

(Total for Question 6 is 9 marks)

The diagram shows a sketch of the curve y = h(x).

Figure (to be drawn): Sketch of the curve y = h(x): a single smooth curve with one minimum turning point at (1, -2), marked and labelled; the curve crosses the x-axis at the single labelled point (4, 0); the curve is decreasing for x < 1 and increasing for x > 1; no other points are labelled.

(a)State the coordinates of the turning point of the curve y = -h(x), and state whether it is a maximum or a minimum.(2)

(b)State the root of the curve y = h(-x).(2)

(c)State the coordinates of the turning point of the curve y = -h(-x), and state whether it is a maximum or a minimum.(3)

(Total for Question 7 is 7 marks)

The diagram shows a sketch of the curve y = p(x). The curve y = q(x) is obtained from the curve y = p(x) by a stretch, scale factor 1/2, parallel to the x-axis, followed by a translation of 3 units in the positive y-direction.

Figure (to be drawn): Sketch of the curve y = p(x): a single smooth curve with one maximum turning point at (3, 4), marked and labelled; the curve passes through the origin (0, 0), labelled; the curve is increasing for x < 3 and decreasing for x > 3; no other points are labelled.

(a)Write down q(x) in terms of p.(2)

(b)Find the coordinates of the image of the maximum turning point of y = p(x) under this combined transformation.(3)

(c)Find the coordinates of the image of the point (0, 0) on y = p(x) under the same combined transformation.(2)

(d)A different combined transformation is applied to y = p(x): a translation of 2 units in the negative x-direction, followed by a stretch, scale factor 3, parallel to the y-axis. State the resulting equation in terms of p, and find the image of the maximum turning point.(3)

(Total for Question 8 is 10 marks)

The point P(2, 5) lies on the curve y = k(x). The curve y = m(x) is defined by m(x) = -2*k(x - 1) + 3.

(a)Describe fully the sequence of three geometric transformations that maps the curve y = k(x) onto the curve y = m(x), stating the order in which they should be applied.(4)

(b)Find the coordinates of the image of the point P under this combined transformation.(3)

(c)Given also that the curve y = k(x) passes through the point Q(-3, 0), find the coordinates of the image of Q under the same combined transformation.(3)

(Total for Question 9 is 10 marks)

The diagram shows a sketch of the curve y = f(x) for -4 ≤ x ≤ 6.

Figure (to be drawn): Sketch of the curve y = f(x) for -4 ≤ x ≤ 6: the curve crosses the x-axis at the two labelled points (-2, 0) and (4, 0); between these two points the curve dips below the x-axis to a single minimum turning point at (1, -6), labelled; for x < -2 and for x > 4 the curve lies above the x-axis (f(x) > 0) and rises away from the axis with no further labelled features; there are no other turning points.

(a)On a sketch of y = |f(x)|, show the effect of the modulus transformation. State the coordinates of the images of the minimum turning point and the two x-intercepts.(4)

(b)By considering the interval -2 ≤ x ≤ 4 only, state the number of solutions to the equation |f(x)| = 3 in this interval, giving a reason for your answer.(2)

(c)State the number of solutions to the equation |f(x)| = -2, and justify your answer briefly.(2)

(Total for Question 10 is 8 marks)

The diagram shows a sketch of the curve y = g(x) for -4 ≤ x ≤ 6.

Figure (to be drawn): Sketch of the curve y = g(x) for -4 ≤ x ≤ 6: for x ≥ 0, the curve passes through the labelled point (0, 2), has a single minimum turning point at (3, -4), labelled, and crosses the x-axis at the labelled point (5, 0); for x < 0, the curve passes through the labelled points (-2, 5) and (-4, 0); no other points are labelled.

(a)On a sketch of y = g(|x|), show the curve for -6 ≤ x ≤ 6. State the coordinates of all labelled points on your sketch.(4)

(b)Explain why the curve y = g(|x|) is always symmetric about the y-axis, whatever the original function g.(2)

(Total for Question 11 is 6 marks)

The curve C has equation y = 1/(x - 2), for x not equal to 2.

(a)Sketch the curve C, labelling the equations of both asymptotes.(3)

(b)The curve D has equation y = |1/(x - 2)|. Sketch the curve D for -2 ≤ x ≤ 6, x not equal to 2, and state the equations of its asymptotes.(3)

(c)The curve E is obtained from C by a stretch, scale factor 1/3, parallel to the y-axis, followed by a translation of 1 unit in the negative x-direction. Find the equation of E in the form y = 1/(ax + b), where a and b are integers to be found, and state the equation of its vertical asymptote.(4)

(Total for Question 12 is 10 marks)

Mark scheme · P17 Pure: Functions, Graphs and Transformations

Question 1

(a) B1 at least two of (-3, 0), (1, 0), (2, 0) stated correctly
(a) B1 all three x-intercepts correct: (-3, 0), (1, 0), (2, 0), cao
(a) Answer: (-3, 0), (1, 0), (2, 0)
(b) B1 (0, 6) cao
(b) Answer: (0, 6)
(c) B1 y -> +infinity as x -> +infinity, and y -> -infinity as x -> -infinity (since the highest power of x, x³, has a positive coefficient), both directions needed
(c) Answer: y -> +infinity as x -> +infinity; y -> -infinity as x -> -infinity
(d) B1 correct general shape: single curve with a local maximum between x = -3 and x = 1 and a local minimum between x = 1 and x = 2, consistent with (c), ft
(d) B1 all three x-intercepts labelled (-3, 0), (1, 0), (2, 0), ft from (a)
(d) B1 y-intercept labelled (0, 6), ft from (b)
(d) Answer: Cubic curve rising from bottom-left, local maximum between x=-3 and x=1, local minimum between x=1 and x=2, then rising to top-right; passes through (-3,0), (1,0), (2,0) and (0,6)

Question 2

(a) B1 x = 0 cao
(a) B1 y = 0 cao
(a) Answer: x = 0 and y = 0
(b) B1 correct branch for x > 0: positive, decreasing towards y=0 as x increases, increasing without bound as x -> 0 from the right
(b) B1 correct branch for x < 0: negative, increasing towards y=0 as x decreases, decreasing without bound as x -> 0 from the left
(b) B1 both asymptotes x=0 and y=0 drawn/labelled
(b) Answer: Two-branch curve: one branch in the region x>0, y>0 approaching y=0 as x increases and approaching the y-axis as x->0+; one branch in the region x<0, y<0 approaching y=0 as x decreases and approaching the y-axis as x->0-; asymptotes x=0 and y=0
(c) B1 both branches of C2 lie above the x-axis (y > 0 for all x not equal to 0), since squaring removes negative values, unlike C1 which is negative for x < 0
(c) B1 C2 is symmetric about the y-axis (an even function), whereas C1 has rotational symmetry about the origin (an odd function)
(c) Answer: C2 lies entirely above the x-axis and is symmetric about the y-axis; C1 has one branch above the axis (x>0) and one below (x<0), with rotational symmetry about the origin
(d) B1 asymptotes x = 0 and y = 0, both needed
(d) B1 C2 crosses neither axis: 0 intercepts, cao
(d) Answer: Asymptotes x = 0 and y = 0; C2 crosses neither axis (0 intercepts)

Question 3

(a) M1 correctly states the proportional relationship C = kn (oe)
(a) A1 k = 15 cao
(a) Answer: C = 15n, k = 15
(b) B1 105 pounds, ft using k from (a)
(b) Answer: 105 pounds
(c) B1 straight line through the origin, drawn only for n > 0
(c) B1 gradient labelled as 15, ft from (a)
(c) B1 coordinates of one further point on the line correct, e.g. (4, 60), ft
(c) Answer: Straight line through the origin, gradient 15, e.g. passing through (4, 60)
(d) M1 correctly states the proportional relationship F = k/w (oe)
(d) A1 k = 480, so F = 480/w, cao
(d) B1 correct branch shape: single decreasing curve lying entirely in the region w > 0, F > 0, never crossing either axis
(d) B1 asymptotes labelled w = 0 and F = 0
(d) B1 one correct point on the curve labelled, e.g. (3, 160), ft
(d) Answer: F = 480/w; asymptotes w = 0 and F = 0; e.g. passes through (3, 160)

Question 4

(a) M1 correctly states the proportional relationship h = kt² (oe)
(a) A1 k = 5 cao
(a) Answer: h = 5t², k = 5
(b) M1 substitutes t = 3.5 into h = 5t²
(b) A1 h = 61.25 m cao
(b) Answer: 61.25 m
(c) B1 correct shape: single increasing, concave-up curve (positive quadratic through the origin), no other turning points
(c) B1 endpoint (0, 0) labelled
(c) B1 endpoint (4, 80) labelled, ft from (a)
(c) Answer: Increasing, concave-up curve from (0, 0) to (4, 80)
(d) M1 correctly states the proportional relationship I = k/x² (oe)
(d) A1 k = 180, so I = 180/x², cao
(d) M1 sets 5 = 180/x² and rearranges to x² = 36
(d) A1 x = 6 m cao (rejects x = -6 since distance must be positive)
(d) Answer: I = 180/x²; x = 6 m
(e) B1 y is multiplied by 4, oe
(e) B1 y is multiplied by √2, oe
(e) Answer: y proportional to x²: multiplied by 4. y proportional to √x: multiplied by √2
(f) M1 correctly states the proportional relationship T = k*√L (oe)
(f) A1 k = 2, so T = 2*√L, cao
(f) B1 T = 6 seconds, ft using k from this part
(f) Answer: T = 2*√L; T = 6 seconds when L = 9

Question 5

(a) M1 recognises the vertical translation by 4 units: y-coordinate of the minimum increases by 4, x-coordinate unchanged
(a) A1 (2, 1) cao
(a) Answer: (2, 1)
(b) M1 recognises the horizontal translation by 3 units in the positive x-direction: x-coordinate of the minimum increases by 3, y-coordinate unchanged
(b) A1 (5, -3) cao
(b) Answer: (5, -3)
(c) M1 recognises a translation of 1 unit in the negative x-direction combined with a translation of 2 units in the negative y-direction
(c) A1 (-1, 3) cao
(c) Answer: (-1, 3)
(d) M1 recognises the horizontal translation by 3 units in the positive x-direction applies to every point on the curve, including the x-intercepts
(d) A1 x = a + 3
(d) A1 x = b + 3, both cao
(d) Answer: x = a + 3 and x = b + 3

Question 6

(a) M1 recognises a vertical stretch, scale factor 3: y-coordinate is multiplied by 3, x-coordinate unchanged
(a) A1 (-1, 18) cao
(a) Answer: (-1, 18)
(b) M1 recognises a horizontal stretch, scale factor 1/2: x-coordinate is multiplied by 1/2, y-coordinate unchanged
(b) A1 (-0.5, 6) cao
(b) Answer: (-0.5, 6)
(c) M1 recognises a horizontal stretch, scale factor 2: x-coordinate is multiplied by 2, y-coordinate unchanged
(c) A1 (-6, 0)
(c) A1 (4, 0), both cao
(c) Answer: (-6, 0) and (4, 0)
(d) M1 maximum value = 2(6) - 5 = 7
(d) B1 occurs at x = -1 cao (the location of the maximiser is unchanged, since 2g(x) - 5 is a strictly increasing function of g(x))
(d) Answer: Maximum value 7, occurring at x = -1

Question 7

(a) M1 recognises a reflection in the x-axis: y-coordinate is negated, x-coordinate unchanged, and a minimum becomes a maximum
(a) A1 (1, 2), a maximum, both needed, cao
(a) Answer: (1, 2), a maximum
(b) M1 recognises a reflection in the y-axis: x-coordinate is negated, y-coordinate unchanged
(b) A1 x = -4 cao
(b) Answer: x = -4
(c) M1 recognises the combined effect: reflecting in the y-axis (x -> -x) and reflecting in the x-axis (y -> -y), applied to the point (1, -2)
(c) A1 (-1, 2) cao
(c) A1 a maximum, cao (reflecting a minimum in the x-axis produces a maximum; reflecting in the y-axis does not change whether a point is a maximum or minimum)
(c) Answer: (-1, 2), a maximum

Question 8

(a) M1 correctly applies the horizontal stretch, scale factor 1/2, first: replaces x with 2x to give p(2x)
(a) A1 q(x) = p(2x) + 3 cao
(a) Answer: q(x) = p(2x) + 3
(b) M1 applies the horizontal stretch to the x-coordinate: 3 * (1/2) = 1.5
(b) A1 applies the translation to the y-coordinate: 4 + 3 = 7
(b) A1 (1.5, 7) cao
(b) Answer: (1.5, 7)
(c) M1 applies the horizontal stretch to the x-coordinate: 0 * (1/2) = 0, and the translation to the y-coordinate: 0 + 3 = 3
(c) A1 (0, 3) cao
(c) Answer: (0, 3)
(d) M1 forms the combined function correctly in the given order: translate first to get p(x + 2), then stretch to get r(x) = 3p(x + 2)
(d) A1 correctly translates the x-coordinate of the maximum: 3 - 2 = 1
(d) A1 correctly stretches the y-coordinate: 4 * 3 = 12, giving image (1, 12), cao
(d) Answer: y = 3p(x + 2); image of the maximum turning point = (1, 12)

Question 9

(a) B1 a translation of 1 unit in the positive x-direction (applied first, giving k(x - 1))
(a) B1 a stretch, scale factor -2, parallel to the y-axis (applied second, giving -2k(x - 1)); accept described as a stretch scale factor 2 combined with a reflection in the x-axis, provided both the factor and the reflection are stated
(a) B1 a translation of 3 units in the positive y-direction (applied third and last, giving -2k(x - 1) + 3)
(a) B1 correct order stated: translation in x, then stretch in y, then translation in y, matching this build-up
(a) Answer: Translation of 1 unit in the positive x-direction, then a stretch scale factor -2 parallel to the y-axis, then a translation of 3 units in the positive y-direction, applied in that order
(b) M1 applies the horizontal translation to the x-coordinate: 2 + 1 = 3
(b) M1 applies the stretch, reflection and vertical translation to the y-coordinate: 5*(-2) + 3 = -7
(b) A1 (3, -7) cao
(b) Answer: (3, -7)
(c) M1 applies the horizontal translation to the x-coordinate: -3 + 1 = -2
(c) M1 applies the stretch, reflection and vertical translation to the y-coordinate: 0*(-2) + 3 = 3
(c) A1 (-2, 3) cao
(c) Answer: (-2, 3)

Question 10

(a) B1 the section of the curve that lies below the x-axis (for -2 < x < 4) is reflected in the x-axis; the sections that already lie above the x-axis (x < -2 and x > 4) are unchanged
(a) B1 the minimum turning point (1, -6) becomes a maximum turning point at (1, 6)
(a) B1 the x-intercepts are unchanged at (-2, 0) and (4, 0), since points on the x-axis map to themselves
(a) B1 correct overall shape: the entire curve lies on or above the x-axis, touching (not crossing) the x-axis at x = -2 and x = 4
(a) Answer: y = |f(x)| matches y = f(x) for x ≤ -2 and x ≥ 4; between x = -2 and x = 4 the curve is reflected in the x-axis, giving a maximum turning point at (1, 6); x-intercepts unchanged at (-2, 0) and (4, 0); the entire curve lies on or above the x-axis
(b) B1 2 solutions
(b) B1 correct reason: |f(x)| rises continuously from 0 (at x = -2) to 6 (at x = 1) then falls continuously back to 0 (at x = 4); since 0 < 3 < 6, the horizontal line y = 3 meets this single hump exactly once on each side of x = 1
(b) Answer: 2 solutions
(c) B1 0 solutions (no solutions)
(c) B1 correct justification: |f(x)| ≥ 0 for all x, so it can never equal a negative number
(c) Answer: 0 solutions, since |f(x)| ≥ 0 for every x

Question 11

(a) B1 recognises that the part of the given curve for x < 0 (through (-2, 5) and (-4, 0)) plays no part in y = g(|x|)
(a) B1 the curve for x ≥ 0 is retained exactly as given, through (0, 2), minimum (3, -4), and (5, 0)
(a) B1 the curve for x < 0 is the mirror image, in the y-axis, of the retained x ≥ 0 branch, giving a second minimum at (-3, -4) and a second x-intercept at (-5, 0)
(a) B1 correct overall shape: symmetric about the y-axis, with minima at (3, -4) and (-3, -4), x-intercepts at (5, 0) and (-5, 0), and passing through (0, 2)
(a) Answer: y = g(|x|) is symmetric about the y-axis: it retains the x ≥ 0 branch through (0, 2), minimum (3, -4), (5, 0), and mirrors it for x < 0, giving a second minimum at (-3, -4) and x-intercept at (-5, 0); the given information about g for x < 0 (points (-2, 5) and (-4, 0)) plays no part
(b) B1 states that replacing x with -x leaves |x| unchanged, since |-x| = |x| for every x
(b) B1 correctly links this to g(|x|) being an even function, and every even function having a graph symmetric about the y-axis
(b) Answer: g(|x|) is an even function, because replacing x with -x leaves |x| unchanged (|-x| = |x|), so g(|-x|) = g(|x|) for every x; every even function has a graph symmetric about the y-axis