T Madas T Madas The graph of y
- Slides: 40
© T Madas
© T Madas
The graph of y = f(x ) is shown below. Sketch the graph of: (a) f(x ) – 3 (b) f(x + 3) (c) - f(x ) + 3 7 (d) - f(x – 3) f(x ) y f(x ) – 3 6 5 4 3 2 1 -10 -8 -6 -4 -2 -1 x 2 4 6 8 10 -2 -3 -4 -5 -6 -7 © T Madas
The graph of y = f(x ) is shown below. Sketch the graph of: (a) f(x ) – 3 (b) f(x + 3) (c) - f(x ) + 3 7 f(x + 3) (d) - f(x – 3) f(x ) y 6 5 4 3 2 1 -10 -8 -6 -4 -2 -1 x 2 4 6 8 10 -2 -3 -4 -5 -6 -7 © T Madas
The graph of y = f(x ) is shown below. Sketch the graph of: (a) f(x ) – 3 (b) f(x + 3) (c) - f(x ) + 3 7 (d) - f(x – 3) f(x ) y 6 5 4 3 2 1 -10 -8 -6 -4 -2 -1 -2 -3 x 2 4 6 8 10 - f(x ) + 3 -4 -5 -6 -7 - f(x ) © T Madas
The graph of y = f(x ) is shown below. Sketch the graph of: (a) f(x ) – 3 (b) f(x + 3) (c) - f(x ) + 3 7 (d) - f(x – 3) f(x ) y 6 5 4 3 2 1 -10 -8 -6 -4 -2 -1 x 2 4 6 8 10 -2 -3 - f(x + 3) -4 -5 -6 -7 - f(x ) © T Madas
© T Madas
The graph of y = f(x ) is shown below. Sketch the graph of: (a) f(x – 5) (b) 2 f(x ) 7 y f(x – 5) f(x ) 6 5 4 3 2 1 -4 -3 -2 x 1 -1 2 3 4 5 6 7 8 -1 -2 © T Madas
The graph of y = f(x ) is shown below. Sketch the graph of: (a) f(x – 5) (b) 2 f(x ) 7 y 2 f(x ) 6 5 4 3 2 1 -4 -3 -2 x 1 -1 2 3 4 5 6 7 8 -1 -2 © T Madas
© T Madas
The curve with equation y = x 2 has been translated to give the curve with equation y = f(x ) shown in the graph below. Write the equation of f(x ) in terms of x. y x 2 (x – 3)2 4 3 2 1 -3 -2 1 -1 -1 -2 2 3 4 5 6 7 x f(x ) (x – 3)2 – 1 x 2 – 6 x + 8 © T Madas
© T Madas
The curve with equation y = x 2 is shown below. Sketch the curve with equation y = (x – 4)2. y x 2 (x – 4)2 4 3 2 1 -3 -2 1 -1 2 3 4 5 6 7 x -1 -2 © T Madas
© T Madas
a) b) Factorise the expression x 2 – 2 x – 8 Solve the equation x 2 – 2 x – 8 = 0 The graph of f(x ) = x 2 – 2 x – 8 is sketched below. c) Write down the co ordinates of points A and B. d) Sketch the graph of f(x + 3), marking the co ordinates where the graph meets the x axis y f(x + 3) f(x ) x 2 – 2 x – 8 (1, 0) (-5, 0) A (-2, 0) x B (4, 0) x 2 – 2 x – 8 = 0 x+2 x – 4 =0 c c = x+2 x – 4 -2 x= 4 © T Madas
© T Madas
The graph below shows the hyperbola with equation Plot the hyperbolas with equations: y 7 6 5 4 3 2 1 -10 -8 -6 -4 -2 -1 2 4 6 8 10 x -2 -3 -4 -5 -6 -7 © T Madas
The graph below shows the hyperbola with equation Plot the hyperbolas with equations: y 7 6 5 4 3 2 1 -10 -8 -6 -4 -2 -1 2 4 6 8 10 x -2 -3 -4 -5 -6 -7 © T Madas
The graph below shows the hyperbola with equation Plot the hyperbolas with equations: y 7 6 5 4 3 2 1 -10 -8 -6 -4 -2 -1 2 4 6 8 10 x -2 -3 -4 -5 -6 -7 © T Madas
The graph below shows the hyperbola with equation Plot the hyperbolas with equations: y 7 6 5 4 3 2 1 -10 -8 -6 -4 -2 -1 2 4 6 8 10 x -2 -3 -4 -5 -6 -7 © T Madas
© T Madas
1. 2. Show that x 2 + 6 x can be written in the form (x + p)2 + q Describe how does the graph of y = x 2 + 6 x relate to the graph of y = x 2 + 6 x = x 2 + 2 x 3 x x + 32 – 32 = x +3 2 – 9 x 2 + 6 x ≡ x + 3 y = x 2 + 6 x 2 – 9 is the same as i. e. p = 3, q = -9 y = x +3 2 – 9 take y = x 2 shift 3 units to the left shift 9 units down © T Madas
© T Madas
The graph drawn below shows the curve with equation y = f(x ). The curve crosses the y - axis at the point A (0, 7) and it has its lowest point at B (2, 3). Write down the coordinates of A and B under the transformations: 1. y = f(x + 2) 2. y = f(x ) – 3 3. y = f(2 x ) y 1. A(0, 7) A (0, 7) B (2, 3) 2. 3. (-2, 7) (0, 3) B (2, 3) x © T Madas
The graph drawn below shows the curve with equation y = f(x ). The curve crosses the y - axis at the point A (0, 7) and it has its lowest point at B (2, 3). Write down the coordinates of A and B under the transformations: 1. y = f(x + 2) 2. y = f(x ) – 3 3. y = f(2 x ) y A(0, 7) A (0, 7) B (2, 3) 1. 2. (-2, 7) (0, 4) (0, 3) (2, 0) 3. B (2, 3) x © T Madas
The graph drawn below shows the curve with equation y = f(x ). The curve crosses the y - axis at the point A (0, 7) and it has its lowest point at B (2, 3). Write down the coordinates of A and B under the transformations: 1. y = f(x + 2) 2. y = f(x ) – 3 3. y = f(2 x ) y A(0, 7) A (0, 7) B (2, 3) 1. 2. 3. (-2, 7) (0, 4) (0, 7) (0, 3) (2, 0) (1, 3) B (2, 3) x © T Madas
© T Madas
The graph opposite shows the curve y = f(x ) The points P (-7, 0), Q (0, -4) and R (1, 0) all lie on the curve Write down the co ordinates of these points under the transformations: a) b) c) d) e) y = f(x ) + 4 shift up by 4 y = f(x – 4) shift right by 4 y = -f(x ) x-reflection y = 2 f(x ) y-stretch/factor of 2 y = f(2 x ) x-shrink/factor of 2 y R P x Q P (-7, 0) Q (0, -4) R (1, 0) a) b) c) d) e) (-7, 4) (-3, 0) (-7, 0) (-3. 5, 0) (0, 0) (1, 4) (4, -4) (5, 0) (0, 4) (1, 0) (0, -8) (1, 0) (0, -4) (0. 5, 0) © T Madas
© T Madas
A sketch of the curve with equation y = f(x ) is shown below. The vertex of the curve is at (6, 7). Write the coordinates of the vertex of the curve with equation: (a) f(x ) + 3 (b) f(-x ) (d) (3, 7) (-6, 7) f(3 x) n io f 3 ct o ra or nt ct ly co a f on by x in n is io x ct y a fle t re ou ab (6, 10) (c) n io at sl 3 an y Tr ft b le n io at sl 3 an y Tr p b u y f(x + 3) (2, 7) (6, 7) y = f(x ) O x © T Madas
© T Madas
The graph of y = f(x ) is shown below. Sketch the graphs of: (a) f(x + 2) (b) 2 f(x ) (c) f(2 x ) y f(x + 2) represents … … translation by 2 steps … to the left x © T Madas
The graph of y = f(x ) is shown below. Sketch the graphs of: (a) f(x + 2) (b) 2 f(x ) (c) f(2 x ) y f(x + 2) represents … … translation by 2 steps … to the left x 2 f(x) represents … … stretch … in y direction ONLY … by a factor of 2 © T Madas
The graph of y = f(x ) is shown below. Sketch the graphs of: (a) f(x + 2) (b) 2 f(x ) (c) f(2 x ) y f(x + 2) represents … … translation by 2 steps … to the left x 2 f(x) represents … … stretch … in y direction ONLY … by a factor of 2 f(2 x) represents … … compression … in x direction ONLY … by a factor of 2 © T Madas
© T Madas
The graph of sinx is shown below for x between 0° and 360° 1. Write down the co ordinates of the points A, B and C. 2. Sketch the graph of sin 2 x for x between 0° and 360°. y 1. 2 1 The value of sine and cosine lies between… … -1 and 1 A (90, 1) 0. 8 0. 6 0. 4 0. 2 0 B (180, 0) 90° 180° x 270° -0. 2 360° y = sinx -0. 4 -0. 6 -0. 8 -1 -1. 2 C (270, -1) © T Madas
The graph of sinx is shown below for x between 0° and 360° 1. Write down the co ordinates of the points A, B and C. 2. Sketch the graph of sin 2 x for x between 0° and 360°. y 1. 2 1 If y = sinx is y = f(x ) then y = sin 2 x is… … y = f(2 x ) … i. e. squashed by a factor of 2 in x only. A (90, 1) 0. 8 0. 6 0. 4 0. 2 0 -0. 2 -0. 4 B (180, 0) 90° 180° x 270° y = sin 2 x 360° y = sinx -0. 6 -0. 8 -1 -1. 2 C (270, -1) © T Madas
The graph of sinx is shown below for x between 0° and 360° 1. Write down the co ordinates of the points A, B and C. 2. Sketch the graph of sin 2 x for x between 0° and 360°. y 1. 2 1 If y = sinx is y = f(x ) then y = sin 2 x is… … y = f(2 x ) … i. e. squashed by a factor of 2 in x only. A (45, 1) 0. 8 0. 6 0. 4 0. 2 0 -0. 2 -0. 4 B (90, 0) 90° B (180, 0) 180° x 270° 360° y = sin 2 x -0. 6 -0. 8 -1 -1. 2 C (135, -1) C (270, -1) © T Madas
© T Madas
© T Madas
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