Question 1 (3+2+2+4+2+3 = 16 marks)
Consider the quadratic function in standard form: y = −3x2 + 6x + 9.
(a) Use the quadratic formula to write the function in factored form.
(b) Find the x-intercepts of the graph of the function, if they exist.
(c) Find the y-intercept of the graph of the function.
(d) Write the function in vertex form by Completing the Square, and find the vertex.
(e) Is the vertex a maximum or a minimum? Justify your answer mathematically.
(f) Sketch a graph of the function, showing all important features. Your graph does not need to be to scale.
Question 2 (3+5+6+1+3 = 18 marks)
Consider the function y = −2 sin (2x − π) for −π ≤ x ≤ 3.
(a) State the amplitude, period and phase shift, for this function.
(b) Solve 2 sin (2x π) = 0 for π x 3 to find the horizontal intercepts (x-intercepts) of the function.
(c) Using the properties of trigonometric functions, compute the values of x for which the maximum and the minimum values of the function occur.
(d) State the range of the function as an interval.
(e) Using the information obtained in (a)–(d), draw the graph of y = −2 sin (2x − π) for
−π ≤ x ≤ 3.
Question 3 (2+6+3 = 11 marks)
(a) Show that tan2 θ = 1 if and only if sin2 θ = 1/4.
(b) Use part (a) to find all the values of θ, −π ≤ θ ≤ 2π, for which tan2 θ = 1 . Give your
answers in radians.
(c) Verify the following trigonometric identity: cos4 t − cos2 t = sin4 t − sin2 t
Question 4 (4+2+3+3+4 = 16 marks)
(a) Find the equation of the line passing through the points (2, 1) and (0, 3). Give your answer in standard form Ax + By = C.
(b) Find the slope of a line that is perpendicular to the line obtained in part (a) and give a reason for your answer.
(c) Find the equation of the line that is perpendicular to the line obtained in part (a) and that passes through the point (5, −5). Give your answer in standard form.
(d) Graph (by hand) the two equations obtained in part (a) and part (c) on the same Cartesian axes. Label the coordinates of the point of intersection between these lines.
(e) Solve the system of equations obtained in part (a) and part (c) by using the elimination method. Marks will be awarded for checking your solution.
Question 5 (6 marks)
In the figure below (not drawn in scale), the points A, B, and C represent the locations of three hospitals on the shore of a peninsula. These hospitals use drones to safely send medicines from one hospital to another.
A drone departs from A in direction to C, travels at 35 km per hour, and takes 45 minutes until it arrives at C. Another drone, departing from B to C, traveling at 40 km per hour takes 30 minutes until it arrives at C. The drones’ paths are indicated in the Figure below as segments b (from A to C) and a (from B to C), respectively. The angle ACB (at the tip of the peninsula in the picture) is 85◦. Deduce the distance between the towns A and B and. Express your final answer in 3 decimal places.
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