More Functions Practice [30 marks]
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- Osborne Berry
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1 More Functions Practice [30 marks] Water has a lower boiling point at higher altitudes. The relationship between the boiling point of water (T) and the height above sea level (h) can be described by the model T = h where T is measured in degrees Celsius ( C) and h is measured in metres from sea level. Write down the boiling point of water at sea level. 1a. 100 C (A1) (C1) 1b. Use the model to calculate the boiling point of water at a height of 1.37 km above sea level. \(T = \times \) (A1) Note: Award (A1) for 1370 seen, for substitution of their h into the equation C (95.342) (A1) (C3) Notes: If their h is incorrect award at most (A0)(A0). If their h = 1.37 award at most (A0)(A1)(ft). Water boils at the top of Mt. Everest at 70 C. 1c. Use the model to calculate the height above sea level of Mt. Everest. 70 = h Note: Award for correctly substituted equation. h = 8820 m ( ) (A1) (C2) Note: The answer is 8820 m (or 8.82 km.) units are required.
2 Part of the graph of the quadratic function f is given in the diagram below. On this graph one of the x-intercepts is the point (5, 0). The x-coordinate of the maximum point is 3. The function f is given by f(x) = x 2 + bx + c, where b, c Z 2a. (i) b ; (ii) c.
3 (i) 3 = b 2 Note: Award for correct substitution in formula. OR 1 + b + c = b + c = b = 0 Note: Award for setting up 2 correct simultaneous equations. OR 2x + b = 0 Note: Award for correct derivative of f(x) equated to zero. b = 6 (A1) (C2) (ii) 0 = (5) c c = 5 (A1)(ft) (C1) Note: Follow through from their value for b. Note: Alternatively candidates may answer part (a) using the method below, and not as two separate parts. (x 5)( x + 1) x 2 + 6x 5 (A1) b = 6 c = 5 (A1) (C3) The domain of f is 0 x 6. 2b. Find the range of f. 5 y 4 (A1)(ft)(A1)(ft)(A1) (C3) Notes: Accept [ 5, 4]. Award (A1)(ft) for 5, (A1)(ft) for 4. (A1) for inequalities in the correct direction or brackets with values in the correct order or a clear word statement of the range. Follow through from their part (a).
4 The diagram shows part of the graph of y = 2 x + 3, and its horizontal asymptote. The graph passes through the points (0, a) and (b, 3.5). 3a. (i) a ; (ii) b. [4 marks] 0 (i) Note: Award for correct substitution. = 4 (A1) (C2) b (ii) 3.5 = Note: Award for correct substitution. b = 1 (A1) (C2) [4 marks] 3b. Write down the equation of the horizontal asymptote to this graph. y = 3 (A1)(A1) (C2) Notes: y = constant (other than 3) award (A1)(A0).
5 The function g(x) is defined as g(x) = 16 + k(c x ) where c > 0. The graph of the function g is drawn below on the domain x 0. The graph of g intersects the y-axis at (0, 80). 4a. k. 80 = 16 + k(c 0 ) k = 64 (A1) (C2) The graph passes through the point (2, 48). 4b. c. 48 = (c 2 ) Note: Award for substitution of their k and (2, 48) into the equation for g(x). c = 2 ( 1.41) ( ) (A1)(ft) (C2) Notes: Award (A1)(ft) for c = ± 2. Follow through from their answer to part (a).
6 The graph passes through the point (2, 48). 4c. Write down the equation of the horizontal asymptote to the graph of y = g(x). y = 16 (A1)(A1) (C2) Note: Award (A1) for y = a constant, (A1) for 16. Shiyun bought a car in The value of the car V, in USD, is depreciating according to the exponential model V = t, t 0 where t is the time, in years, that Shiyun has owned the car. Write down the value of the car when Shiyun bought it. 5a USD (A1) (C1) 5b. Calculate the value of the car three years after Shiyun bought it. Give your answer correct to two decimal places USD (A1) (C2) Calculate the time for the car to depreciate to half of its value since Shiyun bought it. 5c = t (A1)(ft) Notes: Award (A1)(ft) for seen. Follow through from their answer to part (a). Award for equating their half value to t. Allow the use of an inequality. OR Graphical method (sketch): (A1)(ft) for y = seen on the sketch. Follow through from their answer to part (a). (A1)(ft) for the exponent model and an indication of their intersection with their horizontal line (A1)(ft) (C3)
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