Transforming Graphs y=f(x) Worksheet with Answers

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8.2 Transforming Graphs y=f(x)

EDEXCEL 1MA1 · Calculator allowed · about 90 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

1

The graph of y = g(x) is obtained from the graph of y = f(x) by a single translation. For each equation, write down the column vector that describes the translation from y = f(x) to y = g(x).

(a)g(x) = f(x) + 6(1)

(b)g(x) = f(x - 4)(1)

(c)g(x) = f(x + 2)(1)

(d)g(x) = f(x) - 5(1)

(Total for Question 1 is 4 marks)

2

The curve C has equation y = f(x). It has a minimum turning point at M(3, -4), crosses the x-axis at the points A(1, 0) and B(5, 0), and crosses the y-axis at the point D(0, 2).

(a)State the coordinates of the point on y = f(x) + 5 that corresponds to M.(1)

(b)State the coordinates of the point on y = f(x) + 5 that corresponds to A.(1)

(c)State the coordinates of the point on y = f(x) - 3 that corresponds to D.(1)

(d)State the coordinates of the point on y = f(x) - 3 that corresponds to B.(1)

(Total for Question 2 is 4 marks)

3

Using the same curve C as in Question 2 (minimum M(3, -4), x-intercepts A(1, 0) and B(5, 0), y-intercept D(0, 2)):

(a)State the coordinates of the point on y = f(x - 2) that corresponds to M.(1)

(b)State the coordinates of the point on y = f(x + 4) that corresponds to A.(1)

(c)The curve y = f(x - k) has a minimum point at (9, -4). Find the value of k.(2)

(Total for Question 3 is 4 marks)

4

The function h is defined by h(x) = x² - 6x + 8.

(a)Find and simplify an expression for h(x - 3).(3)

(b)Hence write down the coordinates of the minimum point of y = h(x - 3).(2)

(Total for Question 4 is 5 marks)

5

Using the same curve C as in Question 2 (minimum M(3, -4), x-intercepts A(1, 0) and B(5, 0), y-intercept D(0, 2)):

(a)State the coordinates of the point on y = -f(x) that corresponds to M.(1)

(b)State the coordinates of the point on y = -f(x) that corresponds to D.(1)

(c)State the coordinates of the point on y = f(-x) that corresponds to M.(1)

(d)State the coordinates of the point on y = f(-x) that corresponds to B.(1)

(e)State which of the points A, B and D are invariant (stay in exactly the same position) under the transformation y = -f(x), and explain why.(2)

(Total for Question 5 is 6 marks)

6

Using the same curve C as in Question 2 (minimum M(3, -4), x-intercepts A(1, 0) and B(5, 0), y-intercept D(0, 2)), the curve y = f(x) is transformed to y = 3f(x).

(a)State the coordinates of the point corresponding to M.(1)

(b)State the coordinates of the point corresponding to D.(1)

(c)A student says: "The x-intercepts of y = 3f(x) are still x = 1 and x = 5." Is the student correct? Give a reason.(2)

(Total for Question 6 is 4 marks)

7

Using the same curve C as in Question 2 (minimum M(3, -4), x-intercepts A(1, 0) and B(5, 0), y-intercept D(0, 2)):

(a)State the coordinates of the point on y = f(2x) that corresponds to M.(1)

(b)State the coordinates of the point on y = f(x/2) that corresponds to A.(1)

(c)The curve y = f(kx) has a point corresponding to B at (2, 0). Find the value of k.(2)

(Total for Question 7 is 4 marks)

8

The graph of y = sin(x) for 0 ≤ x ≤ 360 has a maximum point at (90, 1), a minimum point at (270, -1), and passes through (0, 0), (180, 0) and (360, 0).

(a)State the coordinates of the maximum point of y = sin(x) + 2.(1)

(b)State the coordinates of the minimum point of y = sin(x) + 2.(1)

(c)Write down the range of y = sin(x) + 2.(2)

(Total for Question 8 is 4 marks)

9

The graph of y = cos(x) for 0 ≤ x ≤ 360 has a maximum at (0, 1), a zero at (90, 0), a minimum at (180, -1), a zero at (270, 0), and a maximum at (360, 1).

(a)State the coordinates of the point on y = cos(x - 60) that corresponds to the maximum at (0, 1).(1)

(b)State the coordinates of the point on y = cos(x - 60) that corresponds to the minimum at (180, -1).(1)

(c)State the coordinates of the point on y = cos(x - 60) that corresponds to the zero at (90, 0).(2)

(Total for Question 9 is 4 marks)

10

Answer the following multiple choice questions on graph transformations.

(a)The graph of y = f(x) is transformed to give the graph of y = f(x) + 4. Which of the following correctly describes this transformation?(1)

A) Translation by vector (4, 0)
B) Translation by vector (0, 4)
C) Translation by vector (-4, 0)
D) Translation by vector (0, -4)

(b)The graph of y = g(x) is transformed to give the graph of y = g(x - 5). Which of the following correctly describes this transformation?(1)

A) Translation by vector (5, 0)
B) Translation by vector (-5, 0)
C) Translation by vector (0, 5)
D) Translation by vector (0, -5)

(Total for Question 10 is 2 marks)

11

The function h is defined by h(x) = x² - 8x + 10.

(a)Find h(x) + 9, giving your answer in the form x² + bx + c.(2)

(b)Show that the equation h(x) + 9 = 0 has no real roots.(3)

(Total for Question 11 is 5 marks)

12

The curve y = p(x) has a maximum point at (4, 6).

(a)The curve y = p(x) + k has a maximum point at (4, -2). Find the value of k.(2)

(b)The curve y = p(x - a) has a maximum point at (10, 6). Find the value of a.(2)

(Total for Question 12 is 4 marks)

13

Using the same curve C as in Question 2 (minimum M(3, -4), x-intercepts A(1, 0) and B(5, 0), y-intercept D(0, 2)):

(a)State the coordinates of the point on y = f(x + 2) - 5 that corresponds to M.(2)

(b)State the coordinates of the point on y = f(x + 2) - 5 that corresponds to D.(2)

(Total for Question 13 is 4 marks)

14

Transformations can be described using column vectors.

(a)The graph of y = f(x) is translated by the vector (4, -7) to give the graph of y = g(x). Write down an expression for g(x) in terms of f(x).(2)

(b)The graph of y = f(x) is translated to give y = f(x + 3) + 2. Write down the column vector of this translation.(2)

(Total for Question 14 is 4 marks)

15

The curve y = q(x) is given by q(x) = x² + 10x + 30.

(a)Express q(x) in the form (x + a)² + b.(2)

(b)Hence, or otherwise, find the coordinates of the minimum point of the curve y = q(x - 4).(3)

(Total for Question 15 is 5 marks)

16

Using the same curve C as in Question 2 (minimum M(3, -4), x-intercepts A(1, 0) and B(5, 0), y-intercept D(0, 2)), the curve y = f(x) is transformed to y = -2f(x).

(a)State the coordinates of the point corresponding to M.(1)

(b)State the coordinates of the point corresponding to D.(1)

(c)Explain why A and B remain unchanged under this transformation, but M and D do not.(2)

(Total for Question 16 is 4 marks)

17

A curve y = r(x) passes through the point (8, 3).

(a)After the transformation y = r(x) + c, the image of this point is (8, -5). Find the value of c.(2)

(b)The curve y = r(kx) has an image point (2, 3) that corresponds to (8, 3) on y = r(x). Find the value of k.(2)

(Total for Question 17 is 4 marks)

18

The curve y = 1/x has asymptotes x = 0 and y = 0.

(a)State the equations of the asymptotes of the curve y = 1/(x - 3) + 2.(3)

(b)Hence write down the coordinates of the point on y = 1/(x - 3) + 2 that corresponds to the point (1, 1) on y = 1/x.(2)

(Total for Question 18 is 5 marks)

19

The curve y = f(x) has a line of symmetry x = 3.

(a)State the equation of the line of symmetry of the curve y = f(x - 2).(2)

(b)State the equation of the line of symmetry of the curve y = f(2x).(2)

(c)Explain why the curve y = f(x) + 7 has the same line of symmetry, x = 3, as the original curve.(1)

(Total for Question 19 is 5 marks)

20

The curve y = f(x) has a minimum point at (-2, 1). The curve y = g(x), where g(x) = f(x + a), has a minimum point at (3, 1). The curve y = h(x), where h(x) = f(x) + b, has a minimum point at (-2, 8).

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

(b)Find the value of b.(2)

(c)Using your values of a and b, write down the coordinates of the minimum point of the curve y = f(x + a) + b.(2)

(Total for Question 20 is 6 marks)

Mark scheme · 8.2 Transforming Graphs y=f(x)

Question 1

(a) B1 (0, 6) oe
(a) Answer: (0, 6)
(b) B1 (4, 0) oe
(b) Answer: (4, 0)
(c) B1 (-2, 0) oe
(c) Answer: (-2, 0)
(d) B1 (0, -5) oe
(d) Answer: (0, -5)

Question 2

(a) B1 (3, 1) cao
(a) Answer: (3, 1)
(b) B1 (1, 5) cao
(b) Answer: (1, 5)
(c) B1 (0, -1) cao
(c) Answer: (0, -1)
(d) B1 (5, -3) cao
(d) Answer: (5, -3)

Question 3

(a) B1 (5, -4) cao
(a) Answer: (5, -4)
(b) B1 (-3, 0) cao
(b) Answer: (-3, 0)
(c) M1 3 + k = 9 oe
(c) A1 k = 6 cao
(c) Answer: k = 6

Question 4

(a) M1 substitutes (x - 3) for x: (x-3)² - 6(x-3) + 8
(a) M1 expands correctly: x² - 6x + 9 - 6x + 18 + 8
(a) A1 x² - 12x + 35 oe, simplified
(a) Answer: x² - 12x + 35
(b) M1 identifies minimum of h(x) at (3, -1) or completes the square of the part (a) answer as (x-6)² - 1
(b) A1 (6, -1) cao
(b) Answer: (6, -1)

Question 5

(a) B1 (3, 4) cao
(a) Answer: (3, 4)
(b) B1 (0, -2) cao
(b) Answer: (0, -2)
(c) B1 (-3, -4) cao
(c) Answer: (-3, -4)
(d) B1 (-5, 0) cao
(d) Answer: (-5, 0)
(e) B1 identifies A and B
(e) B1 correct reason: both have y = 0, and -0 = 0, so their position is unchanged oe
(e) Answer: A and B

Question 6

(a) B1 (3, -12) cao
(a) Answer: (3, -12)
(b) B1 (0, 6) cao
(b) Answer: (0, 6)
(c) B1 Yes, correct
(c) B1 reason: 3 x 0 = 0, so points where y = 0 are unchanged oe
(c) Answer: Yes, correct

Question 7

(a) B1 (1.5, -4) oe cao
(a) Answer: (1.5, -4)
(b) B1 (2, 0) cao
(b) Answer: (2, 0)
(c) M1 5 / k = 2 oe
(c) A1 k = 2.5 oe
(c) Answer: k = 2.5

Question 8

(a) B1 (90, 3) cao
(a) Answer: (90, 3)
(b) B1 (270, 1) cao
(b) Answer: (270, 1)
(c) M1 -1 ≤ sin(x) ≤ 1 shifted by 2 oe
(c) A1 1 ≤ y ≤ 3 oe
(c) Answer: 1 ≤ y ≤ 3

Question 9

(a) B1 (60, 1) cao
(a) Answer: (60, 1)
(b) B1 (240, -1) cao
(b) Answer: (240, -1)
(c) M1 90 + 60 oe
(c) A1 (150, 0) cao
(c) Answer: (150, 0)

Question 10

(a) B1 B
(a) Answer: B
(b) B1 A
(b) Answer: A

Question 11

(a) M1 adds 9 to the constant term: x² - 8x + 10 + 9
(a) A1 x² - 8x + 19 cao
(a) Answer: x² - 8x + 19
(b) M1 writes x² - 8x + 19 = 0 and identifies a = 1, b = -8, c = 19
(b) M1 correct discriminant calculation: (-8)² - 4(1)(19) = 64 - 76 = -12
(b) A1 cso: since discriminant = -12 < 0, the equation has no real roots
(b) Answer: Discriminant = -12, no real roots

Question 12

(a) M1 6 + k = -2 oe
(a) A1 k = -8 cao
(a) Answer: k = -8
(b) M1 4 + a = 10 oe
(b) A1 a = 6 cao
(b) Answer: a = 6

Question 13

(a) M1 applies one transformation correctly: x = 3 - 2 = 1 or y = -4 - 5 = -9
(a) A1 (1, -9) cao
(a) Answer: (1, -9)
(b) M1 applies one transformation correctly: x = 0 - 2 = -2 or y = 2 - 5 = -3
(b) A1 (-2, -3) cao
(b) Answer: (-2, -3)

Question 14

(a) M1 horizontal shift of 4 gives f(x - 4)
(a) A1 g(x) = f(x - 4) - 7 oe
(a) Answer: g(x) = f(x - 4) - 7
(b) M1 identifies one correct component: -3 or 2
(b) A1 (-3, 2) cao
(b) Answer: (-3, 2)

Question 15

(a) M1 (x + 5)² seen or correct method to complete the square
(a) A1 (x + 5)² + 5 cao
(a) Answer: (x + 5)² + 5
(b) M1 identifies minimum of q(x) as (-5, 5)
(b) M1 applies horizontal shift of +4: x = -5 + 4
(b) A1 (-1, 5) cao
(b) Answer: (-1, 5)

Question 16

(a) B1 (3, 8) cao
(a) Answer: (3, 8)
(b) B1 (0, -4) cao
(b) Answer: (0, -4)
(c) B1 A and B have y = 0, and -2 x 0 = 0, so they are invariant oe
(c) B1 M and D have non-zero y-coordinates, so multiplying by -2 changes their position oe
(c) Answer: A, B invariant since y = 0; M, D move since y not equal to 0

Question 17

(a) M1 3 + c = -5 oe
(a) A1 c = -8 cao
(a) Answer: c = -8
(b) M1 8 / k = 2 oe
(b) A1 k = 4 cao
(b) Answer: k = 4

Question 18

(a) M1 vertical asymptote shifts right 3: x = 3
(a) M1 horizontal asymptote shifts up 2: y = 2
(a) A1 x = 3 and y = 2 both stated cao
(a) Answer: x = 3 and y = 2
(b) M1 x = 1 + 3 oe
(b) A1 (4, 3) cao
(b) Answer: (4, 3)

Question 19

(a) M1 3 + 2 oe
(a) A1 x = 5 cao
(a) Answer: x = 5
(b) M1 3 / 2 oe
(b) A1 x = 1.5 cao
(b) Answer: x = 1.5
(c) B1 a vertical translation moves every point up but does not change any x-coordinate, so the line of symmetry is unaffected oe
(c) Answer: Vertical translation does not change x-coordinates

Question 20

(a) M1 -2 - a = 3 oe
(a) A1 a = -5 cao
(a) Answer: a = -5
(b) M1 1 + b = 8 oe
(b) A1 b = 7 cao
(b) Answer: b = 7
(c) M1 applies both shifts to (-2, 1): x = -2 - (-5), y = 1 + 7
(c) A1 (3, 8) cao
(c) Answer: (3, 8)

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