HAEF IB - MATH HL TEST 2 FUNCTIONS – POLYNOMIALS Date: 14 November 2017 by Christos Nikolaidis Paper 1: Without GDC Marks: /40 Name:____________________________________ Questions 1. [Maximum mark: 5] Consider the function f ( x) = x 4 − 5 x 2 + x + 4 The graph of f is translated by 1 unit to the right and 4 units down. Fund the equation of the translated graph in the form y = ax 4 + bx 3 + cx 2 + dx + e ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... 1 2. [Maximum mark: 7] Consider the functions f ( x) = x 4 − 5 x 2 + x + 4 g ( x) = x−4 (a) Find f o g [2 marks] (b) Find g −1 and state its domain and its range. [3 marks] (c) Find the function h , given that g o h = f [do not simplify the answer] [2 marks] ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... 2 3. [Maximum mark: 7] Consider the polynomial f ( x) = x 4 + ax 2 + x + 4 Find (a) the value of a, if f (x ) leaves a remainder – 1 when divided by x + 1 [3 marks] (b) the sum and the product of the roots of f (x ) [2 marks] (c) the sum of the roots of the polynomial f ( x − 1) [2 marks] ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... 3 4. [Maximum mark: 8] Consider the function f ( x) = x 4 − 5 x 2 + x + 4 , − 2 ≤ x ≤ 1 The endpoints of the graph are A(-2,-2) and B(1,1). Sketch the graphs of the following functions and indicate the coordinates of their endpoints. x (a) y = f 2 . [2 marks] 4 (b) y = f ( x ) + 1 . [3 marks] (c) y = f ( x − 1 ) . [3 marks] 5 5. [Maximum mark: 6] Consider the function f ( x) = x 4 − 5 x 2 + x + 4 and the function g given by x g (x ) 1 2 2 5 3 1 4 3 5 4 (a) Find ( f o g )(1) [2 marks] (b) Find ( g −1 o f )(1) [2 marks] (c) Find a solution of the equation ( g o f )( x ) = 3 [2 marks] ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... 6 6. [Maximum mark: 7] Consider the function f ( x) = x 4 − 5 x 2 + x + 4 , It is given that f −1 x ≤ −1.6 exists. (a) Solve the equation f ( x) = f −1 ( x) [4 marks] (b) Find ( f −1 o f )(−10) [1 mark] (c) Find an expression of the function g , given that ( f −1 o g )( x) = x 3 [2 marks] ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... 7 Paper 2: With GDC Marks: /40 Name:____________________________________ Questions 1. [Maximum mark: 5] Write down the largest possible domain and the range of the following functions Function Domain Range f ( x) = x 2 − 4 f ( x) = x 2 − 4 f ( x) = x 2 + 4 f ( x) = 1 x +1 f ( x) = 4−x 4+ x 2 ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... 1 2. [Maximum mark: 8] Consider the function f ( x) = (2 x − 7)( x + 1) ( x − 2)( x − 3) (a) Sketch the graph of the function by indicating any asymptotes and intersections with the two axes. [5 marks] (b) Write down the domain and the range of the function. [3 marks] ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... 2 3. [Maximum mark: 8] Consider the function f ( x) = x 4 − 5 x 2 + x + 4 , −1 ≤ x ≤ 2 (a) Sketch the graph of the function on the diagram below [2 marks] (b) Solve the inequality f ( x) ≤ 0 [2 marks] (c) Find the range of f . [2 marks] (d) Find the possible values of k, given that the equation f ( x ) = k has exactly 3 solutions [2 marks] ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... 3 4. [Maximum mark: 6] Find the cubic polynomial P (x) given that • • • The sum of its roots is 2 The product of the roots is -18 P (x) is divided by x − 3 • P (x) leaves a remainder -48 when divided by x + 1 ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... 4 5. [Maximum mark: 7] Consider the function g ( x) = x 4 − 2 x 2 + 2 (a) Investigate whether g is even, odd or neither. Justify your answer. [2 marks] (b) The function g can be restricted in a suitable domain of the form x ∈ [ a,+∞ ) so that g −1 exists. Find (i) the minimum possible value of a. (ii) an expression for g −1 . [5 marks] ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... 5 6. [Maximum mark: 6] Find the coordinates of any point on the graph of y = x 4 − 5 x 2 + x + 4 whose distance from the point (5,-2) is 5 units. ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... 6
