Honors Math 4 Final Exam 2016 Lexington High School Mathematics Department

Transcription

1 Name Teacher (circle): Runge Tracy Verner Class block (circle): A C D F G H Honors Math 4 Final Exam 2016 Lexington High School Mathematics Department This is a 90-minute exam, but you will be allowed to work for up to 120 minutes. The exam has 3 parts. Directions for each part appear below. In total, there are 56 points that you can earn. A letter grade scale will be set by the course faculty after the tests have been graded. Part A. Short Problems 4 questions, 2 points each, 8 points total If your answer is correct, you will receive full credit. Showing work is not required. If your answer is incorrect, you may receive half credit if you have shown some correct work. A good pace on this part would be to spend 2-4 minutes per problem. Part B. Medium Problems 6 problems, 4 points each, 24 points total Write a complete, clearly explained solution to each problem. Partial credit will be given. A good pace on this part would be to spend 4-6 minutes per problem. Part C. Long Problems 3 problems, 8 points each, 24 points total Write a complete, clearly explained solution to each problem. Partial credit will be given. A good pace on this part would be to spend 8-12 minutes per problem.

2 Honors Math Final Exam page 2 of 10 Part A. Short Problems 4 problems, 2 points each, 8 points total 1. Consider ΔABC with the following information: AB = 4, BC = 6, and AC = 8. Find the measure of A. Express your final answer in degrees rounded to the nearest hundredth. 2. Suppose f (x) = 4x 7 3x + 9. Find the domain and range of f 1 (x).

3 Honors Math Final Exam page 3 of Find the square roots of 4cis 5π. Express your final answer in a + bi form Find the value of c such that there is at least one solution to this matrix system c

4 Honors Math Final Exam page 4 of 10 Part B. Medium Problems 6 problems, 4 points each, 24 points total 5. Find all solutions to the equation 3 tan 2 ( 5θ ) = sin( 5θ )sec( 5θ ). Your answers should be exact values in radians, not decimal approximations. Also, your answer may not include trig functions or their inverses. 6. Prove the identity sin(2θ) = 2 tanθ. You may assume any identities we proved in class. 1+ tan 2 θ

5 Honors Math Final Exam page 5 of Find the null space of the transformation matrix T = final answer as a vector equation Express your 8. Consider the line (x, y, z) = (2, 2, 4)+ t 3, 2,1. a. Find the Cartesian equation of a plane perpendicular to this line, which goes through the point (7, 7, 7). b. Find the point of intersection between the original line and the plane in part a.

6 Honors Math Final Exam page 6 of Find the values of a and b such that is an eigenvector of the matrix a b 2 a a. Find a sinusoidal function f (θ) with a period of 4π, a maximum occurring at (0, 5) and a minimum f (θ) value of 1. Hint: It may be helpful to sketch a graph of f (θ). b. Sketch the polar graph r = f (θ) on the domain 0 θ 4π. Your sketch must include at least 3 exact points labeled with their coordinates.

7 Honors Math Final Exam page 7 of 10 Part C. Long Problems 3 problems, 8 points each, 24 points total a. List the sequence of transformations that would transform the graph of f (x) = x into the graph of g(x) = ( 2x + 4) 3. b. Find the new parametric equations of the conic section 4x 2 + 9y 2 +16x 54y + 61= 0 after it has been shifted down 5 units, and shifted right 3 units. c. Find the single, exact transformation matrix in 3D that performs the following transformations: First, reflects over the x-axis. Then, reflects over the plane y = z.

8 Honors Math Final Exam page 8 of Suppose a quadrilateral ABCD is defined by the points A(3,1, 4,1), B(4, 0, 5, 0), C(5,1, 6,1), and D(4, 2, 5, 2). Note: These four points are coplanar. a. Find the vectors AB, BC, CD, and DA. b. Use the vectors you found in part a to support or refute the claim that quadrilateral ABCD is a square. c. Can the vector w = 1, 5,1, 5 be expressed as a linear combination of AB and BC? If so, write w = k 1 AB ( ) + k 2 ( BC), with the values you find for k 1 and k 2.

9 Honors Math Final Exam page 9 of (Question 13 extends over 2 pages.) Consider a rational function in the form R(x) = P(x) Q(x), where P(x) and Q(x) are polynomials with real coefficients. We know the following information: lim x R(x) = 0.25 P(x) has degree 5. P 2cis π = 0 3 The only x-intercept of R(x) is 2, and it is the minimum of the function. ( ) = Q(x) (x 3) LCM Q(x), x 6 + x 5 14x 4 2x x 2 + x 12 R(x) only has 2 vertical asymptotes. a. Find the function formula of R(x) in the form R(x) = P(x) Q(x).

10 Honors Math Final Exam page 10 of b. Sketch a graph of R(x). Label any intercepts, holes, and asymptotes.