Graphs of Related Functions

The graphs of all functions cut the x-axis where y = 0 Points to Remember The graphs of all functions cut the x-axis where y = 0 The graphs of all functions cut the y-axis where x = 0

Graph of –f(x) Transformations reflects f(x) in the x - axis y = f(x) y = x2 y = – f(x) y = –x2 y = –f(x) reflects f(x) in the x - axis

reflects f(x) in the x - axis y = f(x) y = x3 y = – f(x) y = – x3 y = –f(x) reflects f(x) in the x - axis

Graph of f(– x) Transformations reflects f(x) in the y - axis y = f(x) y = (x – 2)2 Happy parabola x = 0  y = (–2)2 = 4 y = 0  (x – 2)2 = 0  x = 2 (twice) So x-axis tangent y = f(– x ) y = (– x – 2)2 y = f(– x) reflects f(x) in the y - axis

Graph of f(x) ± k Transformations Depending on the value of k y = f(x) y = x2 y = f(x) ± k y = x2 + 3 y = x2 – 1 y = f(x) ± k moves f(x) up or down Depending on the value of k + k  move up – k  move down

Graph of f(x ± k) Transformations y = f(x) y = f(x ± k) y = x2 y = (x – 1)2 y = (x + 2)2 y = f(x ± k) moves f(x) to the left or right depending on the value of k – k  move right + k  move left

Graph of k f(x) Transformations y = f(x) y = x2 – 1 y = k f(x) y = 4(x2 – 1) y = 0.25(x2 – 1 ) y = k f(x) Multiply y coordinate by a factor of k k > 1  (stretch in y-axis direction) 0 < k < 1  (squash in y-axis direction)

Graph of f(kx) Transformations y = f(x) y = f(2x) (- 2 , 10) - 3 - 6 ( - 4, 10) y = f(x) ( - 4, 10) (1,8) (2,8) f(kx) : The value of k scales the graph in x-direction k = 2, x - scale factor = 1/2

Graph of f(kx) Transformations (1,13) (3,8) 8 y = f(x) y = f(1/2x) y = f(x) (3,8) (1,13) 8 (2,13) (6,8) f(kx) : The value of k scales the graph in x-direction k = 1/2, x - scale factor = 2

Graph moves 2 units to the right y = logax (a , 1) (a + 2 , 1) (1 , 0) (3 , 0) y = loga(x – 2) Graph moves 2 units to the right

Find the values of a and b. y = loga(x + b) Find the values of a and b. ( 22 , 2) 3 units left  b = 3 ( –2 , 0) ( 1 , 0) y = loga(x + 3) When x = 22, y = 2 2 = loga(22 + 3) y = log5(x + 3) 2 = loga 25  a = 5

y = abx Find the values of a and b. ( 2 , 48) y = abx When x = 0, y = 3 becomes y = 3bx 3 = a×b0 When x = 2, y = 48 3 = a×1 48 = 3b2 a = 3 16 = b2 ( 0 , 3) b = 4 y = 3 × 4 x

The following questions are on Graphs & Functons Non-calculator questions will be indicated

Reflect across the y axis Now scale by 2 in the y direction The diagram shows the graph of a function f. f has a minimum turning point at (0, -3) and a point of inflexion at (-4, 2). a) sketch the graph of y = f(-x). On the same diagram, sketch the graph of y = 2f(-x) a) Reflect across the y axis b) Now scale by 2 in the y direction

Determine the values of a, b and c The diagram shows a sketch of part of the graph of a trigonometric function Determine the values of a, b and c 2a 1 1 in  2 in 2  a is the amplitude: a = 4 b is the number of waves in 2 b = 2 c is where the wave is centred vertically c = 1

10 - f(x) is positive for -1 < x < 5 a) Express f(x) = x2 – 4x + 5 in the form (x – a)2 + b b) On the same diagram sketch i) the graph of y = f(x) ii) the graph of y = 10 - f(x) c) Find the range of values of x for which 10 - f(x) is positive (2, 9) b) a) c) Solve: 10 - f(x) is positive for -1 < x < 5

Sketch the graph of the derived function f ’. The graph of a function f intersects the x-axis at (–a, 0) and (e, 0) as shown. There is a point of inflexion at (0, b) and a maximum turning point at (c, d). Sketch the graph of the derived function f ’. m is + m is + m is - f(x)

The diagram shows the graphs of two quadratic functions y = f(x) and y = g(x) Both graphs have a minimum turning point at (3, 2). Sketch the graph of y = f ’(x) and on the same diagram sketch the graph of y = g‘(x) y=f(x) y=g(x)

Part of the graph of y = f(x) is shown in the diagram. On separate diagrams sketch the graph of a) y = f(x+1) b) y = – 2f(x) Indicate on each graph the images of O, A, B, C, and D. a) graph moves to the left 1 unit b) graph is reflected in the x axis graph is then scaled 2 units in the y direction

(1,4) (6,-8) 3 7 y = – 2f(x) A(-1,2) 3 7 y = f(x) C(6,4) C(5,4) y = f(x+1) 7 2 -1 -2