CHAPTER Further Functions Solutions Practice Questions 7 5

CHAPTER Further Functions Solutions: Practice Questions 7. 5 07

07 1. (i) Practice Questions 7. 5 The diagram shows the graph of a function k(x). Copy the graph into your copybook. Using the same axes and scale, graph the following: k(x) − 3 Draw a similar line, 3 units below the given line, k(x). (ii) k(x − 3) Draw a similar line, 3 units to the right of the given line, k(x). (iii) k(x + 1) Draw a similar line, 1 unit to the left of the given line, k(x).

07 2. Practice Questions 7. 5 The diagram shows the graph of a function g(x). Copy the graph into your copybook. Using the same axes and scale, graph the following: (i) g(x) – 2: Redraw g(x) graph, 2 units below the given graph. (ii) g(x – 2): Redraw g(x) graph, 2 units to the right of the given graph. (iii) 2 g(x): Draw a graph with the same roots as g(x) but double the height. (iv) −g(x): Draw the image of g(x) graph, as reflected in the x-axis.

07 3. (i) Practice Questions 7. 5 The diagram shows the graph of a function f (x) = x 2 − 4. Copy the graph into your copybook. Using the same axes and scale, graph the following: g (x) = f (x) + 3 moves the original graph f (x) up 3 units. (ii) k(x) = f (x + 1) − 3 k(x) = f (x + 1) – 3: Move 1 unit to the left and down 3 units. (iii) : the curve f (x) will appear to compress (get vertically shorter). It will have the same roots as f (x) but only the height.

07 4. (i) Practice Questions 7. 5 The diagram shows the graph of a cubic function. Copy the graph into your copybook. Using the same axis and scale, graph the following: f (x) + 2: Moves the graph up 2 units. (ii) f (x + 2) − 1 f(x + 2) – 1: Moves the graph 2 units to the left then down 1 unit.

07 5. Practice Questions 7. 5 The diagram shows a graph of the function, f (x) = x 3. The graph is moved as shown: 6 units horizontally to the right 2 units vertically upwards. Write the function of the new graph, in terms of x. f(x) = x 3 Moved 6 units to the right and 2 units up gives: New graph (x – 6)3 + 2

07 6. (i) Practice Questions 7. 5 For each of the following, describe fully the transformation which maps the graph of f (x) onto the graph of g (x). f (x) = 3 x onto g(x) = 3 x + 2 f(x) = 3 x → g(x) = 3 x + 2 2 is added to the entire function, therefore it moves up two units. (ii) f (x) = x 2 onto g(x) = x 2 − 3 f(x) = x 2 → g(x) = x 2 – 3 3 is subtracted from the entire function, therefore it moves down three units.

07 6. (iii) Practice Questions 7. 5 For each of the following, describe fully the transformation which maps the graph of f (x) onto the graph of g (x). f (x) = 4 x 3 onto g (x) = 4(x + 5)3 f(x) = 4 x 3 → g(x) = 4(x + 5)3 5 is added to the x part, therefore it moves the graph five units to the left. (iv) f (x) = 2 x 2 + 3 x + 7 onto g(x) = 2 x 2 + 3 x + 4 f(x) = 2 x 2 + 3 x + 7 → g(x) = 2 x 2 + 3 x + 4 3 is subtracted from the entire function, therefore it moves down three units.

07 6. (v) Practice Questions 7. 5 For each of the following, describe fully the transformation which maps the graph of f (x) onto the graph of g (x). f (x) = 6 x onto g(x) = 6 x − 1 f(x) = 6 x → g(x) = 6 x – 1 1 is subtracted from the x part, therefore it moves the graph 1 unit to the right. (vi) f (x) = 4 x + 3 onto g(x) = 4 x + 2 f(x) = 4 x + 3 → 4 x + 2 2 is added to the x part and 1 is subtracted from the entire function. Therefore, it moves the graph two units to the left then one unit down.

07 7. Practice Questions 7. 5 Function A (blue) is defined by the equation y = x 2. By observation, or otherwise, write down the equation of the functions B, C and D. A: y = x 2 The graph of B is 3 units up above A, so we add 3 to the A function. B: y = x 2 + 3 The graph of C is 2 units to the right of A, so we subtract 2 from the x part within the A function. C: y = (x – 2)2 The graph of D is 3 units to the left and 2 units down from A, so we add 3 to the x part and subtract 2 from the entire A function. D: y = (x + 3)2 – 2

07 8. Practice Questions 7. 5 The graph of y = f (x) is shown in red. Write down the equations of g (x) and h(x) in terms of f (x). g(x) has the same roots as f(x), but is 3 times higher. Therefore, g(x) = 3 f (x). h(x) is two units to the left and two units above f(x), so we add 2 to the x part and also add 2 to the entire function to get: h(x) = f (x + 2) + 2