VIEWS: 135 PAGES: 9 CATEGORY: Education POSTED ON: 3/10/2010
GCSE MATHEMATICS Higher Tier, topic sheet. TRANSFORMATION OF GRAPHS WRITE YOUR ANSWERS ON THESE SHEETS 1. Diagram 1 shows a sketch of y = x². y O x Diagram 1 Diagrams 2, 3 and 4 show sketches of the graph of y = x² after it has been moved parallel to the x-axis, or parallel to the y-axis, or parallel to both. y y y 3 3 O x O 2 x O 2 x Diagram 2 Diagram 3 Diagram 4 Write down the equations of the graphs shown in (a) Diagram 2, (b) Diagram 3, (c) Diagram 4. 2. The diagram shows a sketch of the function y = f (x). y The curve cuts the x-axis at (0, 0) and (3, 0). 6 A (1, 2) is a point on the curve. 4 A (1, 2) 2 0 x –1 1 2 3 4 –2 The function is transformed. –4 On each grid below, sketch the transformed function. State the co-ordinates of A on each new curve. –6 www.mathsguru.co.uk y (a) y = f (x) – 2 6 A is at (……….. , …………) 4 2 0 x –1 1 2 3 4 –2 –4 –6 (b) y = 2f (x) y 6 A is at (……….. , …………) 4 2 0 x –1 1 2 3 4 –2 –4 –6 (c) y = f (2x) y 6 A is at (……….. , …………) 4 2 0 x –1 1 2 3 4 –2 –4 –6 www.mathsguru.co.uk 3. The sketch below is of y = x2. y x O On the axes provided, sketch the following graphs. y = x2 is shown dotted on each set of axes. In each case p is a positive integer greater than 1. (a) Sketch y = px2 y x O (b) (i) Sketch y = x2 + p y x O (ii) Write down the coordinates of the point where the graph crosses the y-axis. (c) (i) Sketch y = (x + p)2 y x O (ii) Write down the coordinates of the point where the graph touches the x-axis. www.mathsguru.co.uk 4. The diagram shows the graph of y = cos x for 0° x 360° y 3 2 1 0 90º 180º 270º 360º x –1 –2 –3 On the axes below sketch the graphs of (a) y = 2 cos x for 0° x 360° (b) y = cos2x for 0° x 360° y y 3 3 2 2 1 1 0 0 90º 180º 270º 360º x 90º 180º 270º 360º x –1 –1 –2 –2 –3 –3 (c) y = 2 + cos x for 0° x 360° y 3 2 1 0 90º 180º 270º 360º x –1 –2 –3 www.mathsguru.co.uk 5. The diagram shows a sketch of the graph of y = f(x) for – 3 x 3 y 5 4 3 2 1 –5 –4 –3 –2 –1 1 2 3 4 5x –1 –2 –3 –4 –5 (a) Draw each of the graphs with these equations. (i) y = f(x) + 2 (ii) y = f(x – 2) y y 5 5 4 4 3 3 2 2 1 1 –5 –4 –3 –2 –1 1 2 3 4 5x –5 –4 –3 –2 –1 1 2 3 4 5x –1 –1 –2 –2 –3 –3 –4 –4 –5 –5 (b) The graph below is a transformation of the graph of y = f(x). Write down, in terms of f(x), the equation of the graph. y 5 4 3 2 1 –5 –4 –3 –2 –1 1 2 3 4 5x –1 –2 –3 –4 y –5 5 6. The diagram shows the graph of y = x2 for 2 x 2. –5 0 5 x –5 www.mathsguru.co.uk Each of the graphs below is a transformation of this graph. Write down the equation of each graph. y y (a) 5 (b) 5 –5 0 5 x –5 0 5 x –5 –5 y c) 5 –5 0 5 x –5 7. The graph of y = sin x for 0° x 360° is shown on the grid below. The point P(90, 1) lies on the curve. y 2 P 1 O 90 180 270 360 x –1 –2 On both of the grids that follow, sketch the graph of the transformed function. In both cases write down the coordinates of the transformed point P. www.mathsguru.co.uk (a) y = sin (x – 45) y 2 1 O 90 180 270 360 x –1 –2 (b) y = 2sinx y 2 1 O 90 180 270 360 x –1 –2 8. This is the graph of y = sin x. y 1 0 0° 90° 180° 270° 360° x –1 On the grid below, sketch the graph of y = sin (x - 90°) www.mathsguru.co.uk y 1 0 0° 90° 180° 270° 360° x –1 9. The graph y = x2 is transformed as shown. y y y = x2 Not drawn accurately O x (–3,0) O x Write down the equation of the transformed graph. www.mathsguru.co.uk Answers. 1. (a) y = x2 + 3. (b) y = (x 2)2. (c) y = (x 2)2 + 3. 2. (a) A = (1, 0). (b) A = (1, 4). (c) A = (0.5, 2). 3. (b) (ii) (0, p). (c) (ii) ( p, 0). 5. (b) y=3 f(x). 6. (a) y = (x + 3)2. (b) y = x2 2. y = ( 1 x) . 2 (c) 2 7. (a) P = (135, 1). (b) P = (90, 2). 9. y = (x + 3)2. www.mathsguru.co.uk