Core Mathematics 1 Transformations of Graphs

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1 Regent College Maths Department Core Mathematics 1 Transformations of Graphs

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3 Transformations of Graphs September 2011 C1 Note Knowledge of the effect of simple transformations on the graph of y f( x) as represented by y af( x), y f( x) a, y f( x a), y f( ax). Students should be able to apply one of these transformations to any of the above functions [quadratics, cubics, reciprocal] and sketch the resulting graph. Given the graph of any function y = f(x) students should be able to sketch the graph resulting from one of these transformations. The following graphs are obtained as shown below: y af( x) by stretching y f( x) by a factor of a parallel to the y-axis. y f( x) a by shifting y f( x) by a in the positive y direction. y f( x a) by shifting y f( x) by a in the negative x direction. y f( ax) by stretching 1 y f( x) by a factor of a parallel to the x-axis. The connection between y f( x) and y f( x 2) is shown below: y f( x 2) y f( x) The connection between y f( x) and y f( x) 3 is shown below: y f( x) 3 y f( x) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 23 Core Maths 1 Transformations of Graphs Page 1

4 The connection between y f( x) and y 2f( x) is shown below: September 2011 C1 Note y 2f( x) y f( x) The connection between y f( x) and y f(2 x) is shown below: y f( x) y f(2 x) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 24 Core Maths 1 Transformations of Graphs Page 2

5 1. y O 2 4 x P(3, 2) shows a sketch of the curve with equation y = f(x). The curve crosses the x-axis at the points (2, 0) and (4, 0). The minimum point on the curve is P(3, 2). In separate diagrams sketch the curve with equation (a) y = f(x), (b) y = f(2x). On each diagram, give the coordinates of the points at which the curve crosses the x-axis, and the coordinates of the image of P under the given transformation. 2. y Jan 2005, Q6 (3, 5) O (6, 0) x Core Maths 1 Transformations of Graphs Page 3

6 shows a sketch of the curve with equation y = f(x). The curve passes through the origin O and through the point (6, 0). The maximum point on the curve is (3, 5). On separate diagrams, sketch the curve with equation (a) y = 3f(x), (b) y = f(x + 2). (2) On each diagram, show clearly the coordinates of the maximum point and of each point at which the curve crosses the x-axis. May 2005, Q4 3. y (0, 3) (4, 0) O (1, 0) x shows a sketch of the curve with equation y = f(x). The curve passes through the points (0, 3) and (4, 0) and touches the x-axis at the point (1, 0). On separate diagrams, sketch the curve with equation (a) y = f(x + 1), (b) y = 2f(x), 1 (c) y = f x. 2 On each diagram show clearly the coordinates of all the points at which the curve meets the axes. Jan 2006, Q6 Core Maths 1 Transformations of Graphs Page 4

7 4. Given that f(x) = x 1, x 0, (a) sketch the graph of y = f(x) + 3 and state the equations of the asymptotes. (b) Find the coordinates of the point where y = f(x) + 3 crosses a coordinate axis. (2) (4) 5. y May 2006, Q3 O x 3 shows a sketch of the curve with equation y =, x 0. x 3 (a) On a separate diagram, sketch the curve with equation y =, x 2, showing the x + 2 coordinates of any point at which the curve crosses a coordinate axis. (b) Write down the equations of the asymptotes of the curve in part (a). (2) May 2007, Q5 Core Maths 1 Transformations of Graphs Page 5

8 6. y (2, 5) 1 4 x shows a sketch of the curve with equation y = f(x). The curve crosses the x-axis at the points (1, 0) and (4, 0). The maximum point on the curve is (2, 5). In separate diagrams, sketch the curves with the following equations. On each diagram show clearly the coordinates of the maximum point and of each point at which the curve crosses the x-axis. (a) y = 2f(x), (b) y = f( x). The maximum point on the curve with equation y = f(x + a) is on the y-axis. (c) Write down the value of the constant a. (1) Jan 2008, Q6 7. Core Maths 1 Transformations of Graphs Page 6

9 shows a sketch of the curve with equation y = f(x). The curve passes through the point (0, 7) and has a minimum point at (7, 0). On separate diagrams, sketch the curve with equation (a) y = f(x) + 3, (b) y = f(2x). (2) On each diagram, show clearly the coordinates of the minimum point and the coordinates of the point at which the curve crosses the y-axis. 8. June 2008, Q3 shows a sketch of the curve C with equation y = f(x). There is a maximum at (0, 0), a minimum at (2, 1) and C passes through (3, 0). On separate diagrams, sketch the curve with equation (a) y = f(x + 3), (b) y = f( x). Core Maths 1 Transformations of Graphs Page 7

10 9. On each diagram show clearly the coordinates of the maximum point, the minimum point and any points of intersection with the x-axis. Jan 2009, Q5 y ( 2, 5) y = f(x) y = 1 O x shows a sketch of part of the curve with equation y = f(x). The curve has a maximum point ( 2, 5) and an asymptote y = 1, as shown in. On separate diagrams, sketch the curve with equation (a) y = f(x) + 2, (b) y = 4f(x), (c) y = f(x + 1). (2) (2) On each diagram, show clearly the coordinates of the maximum point and the equation of the asymptote. Jan 2010, Q8 Core Maths 1 Transformations of Graphs Page 8

11 10. shows a sketch of the curve with equation y = f(x). The curve has a maximum point A at ( 2, 3) and a minimum point B at (3, 5). On separate diagrams sketch the curve with equation (a) y = f (x + 3), (b) y = 2f(x). On each diagram show clearly the coordinates of the maximum and minimum points. The graph of y = f(x) + a has a minimum at (3, 0), where a is a constant. 11. (c) Write down the value of a. (1) May 2010, Q6 Core Maths 1 Transformations of Graphs Page 9

12 shows a sketch of the curve with equation y = f(x) where f(x) = x x 2, x 2. The curve passes through the origin and has two asymptotes, with equations y = 1 and x = 2, as shown in. 12. (a) In the space below, sketch the curve with equation y = f(x 1) and state the equations of the asymptotes of this curve. (b) Find the coordinates of the points where the curve with equation y = f(x 1) crosses the coordinate axes. (4) Jan 2011, Q5 shows a sketch of the curve C with equation y = f(x), where f(x) = x 2 (9 2x). There is a minimum at the origin, a maximum at the point (3, 27) and C cuts the x-axis at the point A. (a) Write down the coordinates of the point A. (1) (b) On separate diagrams sketch the curve with equation (i) y = f(x + 3), (ii) y = f(3x). Core Maths 1 Transformations of Graphs Page 10

13 On each sketch you should indicate clearly the coordinates of the maximum point and any points where the curves cross or meet the coordinate axes. (6) The curve with equation y = f(x) + k, where k is a constant, has a maximum point at (3, 10). 13. (c) Write down the value of k. (1) Jan 2011, Q10 shows a sketch of the curve C with equation y = f(x). The curve C passes through the origin and through (6, 0). The curve C has a minimum at the point (3, 1). On separate diagrams, sketch the curve with equation (a) y = f(2x), (b) y = f(x), (c) y = f(x + p), where p is a constant and 0 < p < 3. (4) On each diagram show the coordinates of any points where the curve intersects the x-axis and of any minimum or maximum points. May 2011, Q8 Core Maths 1 Transformations of Graphs Page 11

14 14. shows a sketch of the curve with equation y = x 2, x 0. The curve C has equation y = x 2 5, x 0, and the line l has equation y = 4x + 2. (a) Sketch and clearly label the graphs of C and l on a single diagram. On your diagram, show clearly the coordinates of the points where C and l cross the coordinate axes. (5) (b) Write down the equations of the asymptotes of the curve C. (2) (c) Find the coordinates of the points of intersection of y = x 2 5 and y = 4x + 2. (5) Jan 2013, Q6 Core Maths 1 Transformations of Graphs Page 12

15 15. shows a sketch of the curve with equation y = f(x) where f(x) = (x + 3) 2 (x 1), x R. The curve crosses the x-axis at (1, 0), touches it at ( 3, 0) and crosses the y-axis at (0, 9). 16. (a) Sketch the curve C with equation y = f(x + 2) and state the coordinates of the points where the curve C meets the x-axis. (b) Write down an equation of the curve C. (c) Use your answer to part (b) to find the coordinates of the point where the curve C meets the y-axis. (2) May 2013, Q8 (1) shows a sketch of the curve C with equation y = f(x). The curve C passes through the point ( 1, 0) and touches the x-axis at the point (2, 0). The curve C has a maximum at the point (0, 4). Core Maths 1 Transformations of Graphs Page 13

16 The equation of the curve C can be written in the form. where a, b and c are integers. y = x 3 + ax 2 + bx + c, (a) Calculate the values of a, b and c. (b) Sketch the curve with equation y = f( 21 x). (5) Show clearly the coordinates of all points where the curve crosses or meets the coordinate axes. May 2013_R, Q9 Core Maths 1 Transformations of Graphs Page 14