Tiêu đề MM3-4 Revision Guides 1-8 Solutions.pdf ✅

Revision Guide 1 - Linear Equations and Transformations Tech-free Question 1 5 marks Let f : [2,∞) → R, f (x) = 2− 9 x +4 . (a) Find the rule and domain of the inverse function of f . [3 marks] Solution: Let y = f (x), swap x and y, then solve for y to get the inverse. y = 2− 9 x +4 x = 2− 9 y +4 (1 mark) 2− x = 9 y +4 y +4 = 9 2− x y = 9 2− x −4 (1 mark) The domain of f −1 should be the same as the range of f , · 1 2 , 2¶ . Therefore, f −1 : · 1 2 , 2¶ → R, f −1 (x) = 9 2− x −4. (1 mark) (b) Sketch the graph of the inverse function of f on the set of axes below. Label any axis intercepts and endpoints with their coordinates and asymptotes with their equations. [2 marks] Mathematical Methods 3/4 © EdAtlas | Page 1 © EdAtlas
Revision Guide 1 - Linear Equations and Transformations Solution: Correct shape (1 mark) Correctly labelled endpoint and asymptote (1 mark) Question 2 2 marks The graph of y = 4x 3 −2x +2 is dilated by a factor of 2 from the y-axis then reflected in the x-axis. (a) Find the rule of the resulting image. [1 mark] Solution: Dilation by a factor of 2 from the y-axis: y1 = 4 3 x 2 ́3 −2 3 x 2 ́ +2 = 1 2 x 3 − x +2 Reflection in the x-axis: y2 = −μ 1 2 x 3 − x +2 ¶ = − 1 2 x 3 + x −2 Therefore, the resulting image is y = − 1 2 x 3 + x −2. (b) If the graph of y = 4x 3 −2x +2 has an x-intercept at (−1, 0), find the coordinates of the x-intercept of the graph in part a). [1 mark] Solution: Only the dilation has an effect on the x-intercept, so the coordinates of the new x-intercept are (−2, 0). Mathematical Methods 3/4 © EdAtlas | Page 2 © EdAtlas
Revision Guide 1 - Linear Equations and Transformations Question 3 2 marks The graph of f : R → R, f (x) = 1 10 (2x +5)(x −1)(x −5) is given below. (a) For what values of m does the graph of f (x−m)have exactly two positive x-intercepts? [1 mark] Solution: For exactly two positive x-intercepts, the graph of f can be moved less than 1 unit in the negative x-direction or at most 5 2 units in the positive x-direction, so m ∈ μ −1, 5 2 , ̧ . (b) For what values of n does the equation f (x)+n = 0 have exactly one solution? [1 mark] Solution: For exactly one solution, the graph of f must be translated more than 245 54 units upward or more than 18 5 units downward, so n ∈ μ −∞,− 18 5 ¶ ∪ μ 245 54 ,∞ ¶ . Mathematical Methods 3/4 © EdAtlas | Page 3 © EdAtlas
Revision Guide 1 - Linear Equations and Transformations Question 4 3 marks Sketch the graph of f : R\ {3} → R, f (x) = 2 (x −3)2 −2. Label all axis intercepts with their coordinates and asymptotes with their equations. Solution: Correct shape (1 mark) Correctly labelled axis intercepts (1 mark) Correctly labelled asymptotes (1 mark) Mathematical Methods 3/4 © EdAtlas | Page 4 © EdAtlas