Final Exam Review Problems

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1 Final Exam Review Problems Name: Date: June 23, 2013 P Determine whether the line x = 4 represens y as a function of x. P Graph f(x) = 3x 1 x 6. Find the x and y-intercepts and asymptotes (if any). Include a table of nice values for which to evaluate f and the corresponding values of f. P Graph f(x) = x(x 2)(x + 3). Find the x and y-intercepts and asymptotes (if any). Include a table of nice values for which to evaluate f and the corresponding values of f.

2 P Graph f(x) = (x 1) Find the x and y-intercepts and asymptotes (if any). Include a table of nice values for which to evaluate f and the corresponding values of f. P Graph f(x) = 4 x Find the x and y-intercepts and asymptotes (if any). Include a table of nice values for which to evaluate f and the corresponding values of f.

3 P Use the graph of f to sketch each graph. (a) y = f(x) + 2 (b) y = f(x 2) (c) y = 2f(x) (d) y = f(x) (e) y = f(x + 3) (f) y = f( x) y 4 4,2 2 3,1 1, x 0, 1 2 (g) y = f ( 1 2 x)

4 P Graph g(x) = 3 (x + 5) 2. Find the x and y-intercepts and asymptotes (if any). Include a table of nice values for which to evaluate f and the corresponding values of f. P Graph g(x) = 3 + 2(x 4) 2 Find the x and y-intercepts and asymptotes (if any). Include a table of nice values for which to evaluate f and the corresponding values of f.

5 P Graph g(x) = 1 2 x 4 Find the x and y-intercepts and asymptotes (if any). Include a table of nice values for which to evaluate f and the corresponding values of f.

6 P Find (a) f g (b) g f and (c) g g. f(x) = 3 x 1 and g(x) = x P A. 1. Simplify the expression. 2(x 2 3) 3 (2x)(x + 1) 3 3(x + 1) 2 (x 2 3) 2 [(x + 1) 3 ] 2

7 P A. 2. Simplify the expression. (6x + 1) 3 (27x 2 + 2) (9x 3 + 2x)(3)(6x + 1) 2 (6) [(6x + 1) 3 ] 2 P A. 3. Simplify the expression. (x + 2) 3/4 (x + 3) 2/3 (x + 3) 1/3 (x + 2) 1/4 [(x + 2) 3/4 ] 2

8 P A. 4. Simplify the expression. 2(3x 1) 1/3 (2x + 1) ( 1 3) (3x 1) 2/3 (3) (3x 1) 2/3 P A. 5. Simplify the expression. 1 (x 2 + 4) 1/2 1 2 (x2 + 4) 1/2 (2x)

9 P A. 6. Simplify the expression. ( ) ( ) 3 1 (x 2 + 5) 1/2 (3x 2) 1/2 (3) + (3x 2) 3/2 (x 2 + 5) 1/2 (2x) 2 2 P CHT A cylindrical can has a volume of 600 cubic centimeters and a radius of 4 centimeters. Find the height of this can. P CHT Graph y = 4 x Find the x and y-intercepts and asymptotes (if any). Include a table of nice values.

10 P CHT Graph y = x 2 1. Find the x and y-intercepts and asymptotes (if any). Include a table of nice values. P CHT Graph f(x) = 2x 6 + 5x 4 x 2. Find the x and y-intercepts and asymptotes (if any). Include a table of nice values for which to evaluate f and the corresponding values of f. Approximate the intervals over which f is increasing, decreasing, or constant. Determine whether the function is even, odd, or neither.

11 P CHT Graph f(x) = 4x 3 x Find the x and y-intercepts and asymptotes (if any). Include a table of nice values for which to evaluate f and the corresponding values of f. Approximate the intervals over which f is increasing, decreasing, or constant. Determine whether the function is even, odd, or neither. P CHT Graph f(x) = x + 5. Find the x and y-intercepts and asymptotes (if any). Include a table of nice values for which to evaluate f and the corresponding values of f. Approximate the intervals over which f is increasing, decreasing, or constant. Determine whether the function is even, odd, or neither.

12 P CHT Graph f(x) = { 3x + 7, 4x 2 1, x 3 x > 3 Find the x and y-intercepts and asymptotes (if any). Include a table of nice values for which to evaluate f and the corresponding values of f. Approximate the intervals over which f is increasing, decreasing, or constant. Determine whether the function is even, odd, or neither. P CHT Graph h(x) = [[x]]. Find the x and y-intercepts and asymptotes (if any). Include a table of nice values for which to evaluate h and the corresponding values of h. Approximate the intervals over which h is increasing, decreasing, or constant. Determine whether the function is even, odd, or neither.

13 P CHT Graph h(x) = x Find the x and y-intercepts and asymptotes (if any). Include a table of nice values for which to evaluate h and the corresponding values of h. Approximate the intervals over which h is increasing, decreasing, or constant. Determine whether the function is even, odd, or neither. P CHT Graph h(x) = 2(x 5) Find the x and y-intercepts and asymptotes (if any). Include a table of nice values for which to evaluate h and the corresponding values of h. Approximate the intervals over which h is increasing, decreasing, or constant. Determine whether the function is even, odd, or neither.

14 P CHT Find (a) (f + g)(x), (b) (f g)(x), (c) (fg)(x), (d) (f/g)(x), (e) (f g)(x), and (f) (g f)(x). f(x) = 3x 2 7, g(x) = x 2 4x + 5 P CHT Find (a) (f + g)(x), (b) (f g)(x), (c) (fg)(x), (d) (f/g)(x), (e) (f g)(x), and (f) (g f)(x). f(x) = 1 x, g(x) = 2 x

15 P CHT Describe how the graph of g differs from the graph of f(x) = x 2. (a) g(x) = 2 x 2 (b) g(x) = ( x 3 2 ) 2 P CHT Find an equation of the parabola. y , , 6 x

16 P CHT The path of a ball is given by y = 1 20 x2 + 3x + 5, where y is the height (in feet) of the ball and x is the horizontal distance (in feet) from where the ball was thrown. (a) Find the maximum height of the ball. (b) Which number determines the height at which the ball was thrown? Does changing this value change the coordinates of the maximum height of the ball? Explain. P CHT Determine the right-hand and left-hand behavior of the graph of the function h(t) = 3 4 t5 + 2t 2. Then sketch the graph. Find the t and y-intercepts and asymptotes (if any). Include a table of nice values for which to evaluate h and the corresponding values of h. Approximate the intervals over which h is increasing, decreasing, or constant. Determine whether the function is even, odd, or neither.

17 P CHT Divide using long division P CHT Divide using synthetic division 3x 3 + 4x + 1 x x 4 5x 2 3 x 2 P CHT Use synthetic division to show that x = 5/2 is a zero of the function given by f(x) = 2x 3 5x 2 6x

18 P CHT Perform each operation and write the result in standard form (a) 10i (3 + 25) (b) (2 + 3i)(2 3i) P CHT Write the quotient in standard form: i. P CHT Find a polynomial function with real coefficients that has the given zeros. 0, 3, 2 + i P CHT Find a polynomial function with real coefficients that has the given zeros. 1 3i, 2, 2

19 P CHT Find all the zeros of f(x) = 3x x 2 7x 10. P CHT Find all the zeros of f(x) = x 4 9x 2 22x + 24.

20 P CHT Determine the right-hand and left-hand behavior of the graph of the function h(x) = 4 1. Then sketch the graph. Find the x and y-intercepts and asymptotes (if any). x 2 Include a table of nice values for which to evaluate h and the corresponding values of h. Approximate the intervals over which h is increasing, decreasing, or constant. Determine whether the function is even, odd, or neither. P CHT Determine the right-hand and left-hand behavior of the graph of the function f(x) = 2x2 5x 12. x 2 16 Then sketch the graph. Find the x and y-intercepts and asymptotes (if any). Include a table of nice values for which to evaluate h and the corresponding values of h. Approximate the intervals over which h is increasing, decreasing, or constant. Determine whether the function is even, odd, or neither.

21 P CHT Determine the right-hand and left-hand behavior of the graph of the function g(x) = x2 + 2 x 1 Then sketch the graph. Find the x and y-intercepts and asymptotes (if any). Include a table of nice values for which to evaluate h and the corresponding values of h. Approximate the intervals over which h is increasing, decreasing, or constant. Determine whether the function is even, odd, or neither. P CHT Solve 2x 2 + 5x > 12. Express your solution in interval notation and sketch a number line corresponding to that interval.

22 P CHT Solve 2 x 1 x + 6 Express your solution in interval notation and sketch a number line corresponding to that interval.

23 P CHT Determine the right-hand and left-hand behavior of the graph of the function f(x) = 10 x Then sketch the graph. Find the x and y-intercepts and asymptotes (if any). Include a table of nice values for which to evaluate f and the corresponding values of f. Approximate the intervals over which f is increasing, decreasing, or constant. Determine whether the function is even, odd, or neither. P CHT Determine the right-hand and left-hand behavior of the graph of the function f(x) = 6 x 2 Then sketch the graph. Find the x and y-intercepts and asymptotes (if any). Include a table of nice values for which to evaluate f and the corresponding values of f. Approximate the intervals over which f is increasing, decreasing, or constant. Determine whether the function is even, odd, or neither.

24 P CHT Determine the right-hand and left-hand behavior of the graph of the function f(x) = 1 e 2x Then sketch the graph. Find the x and y-intercepts and asymptotes (if any). Include a table of nice values for which to evaluate f and the corresponding values of f. Approximate the intervals over which f is increasing, decreasing, or constant. Determine whether the function is even, odd, or neither. P CHT Evaluate (a) log (b) 4.6 ln e 2.

25 P CHT Determine the right-hand and left-hand behavior of the graph of the function f(x) = log x 6 Then sketch the graph. Find the x and y-intercepts and asymptotes (if any). Include a table of nice values for which to evaluate f and the corresponding values of f. Approximate the intervals over which f is increasing, decreasing, or constant. Determine whether the function is even, odd, or neither. P CHT Determine the right-hand and left-hand behavior of the graph of the function f(x) = ln(x 4) Then sketch the graph. Find the x and y-intercepts and asymptotes (if any). Include a table of nice values for which to evaluate f and the corresponding values of f. Approximate the intervals over which f is increasing, decreasing, or constant. Determine whether the function is even, odd, or neither.

26 P CHT Determine the right-hand and left-hand behavior of the graph of the function f(x) = 1 + ln(x + 6) Then sketch the graph. Find the x and y-intercepts and asymptotes (if any). Include a table of nice values for which to evaluate f and the corresponding values of f. Approximate the intervals over which f is increasing, decreasing, or constant. Determine whether the function is even, odd, or neither. P CHT Expand P CHT Expand P CHT Expand log 2 3a 4 ln 5 x 6 log (x 1)3 y 2 z

27 P CHT Condense P CHT Condense P CHT Condense log log 3 y 4 ln x 4 ln y 3 ln x ln(x + 3) + 2 ln y P CHT Solve 5 x = 1 25 P CHT Solve 3e 5x = 132 P CHT Solve e 4x = 5

28 P CHT Solve ln x = 1 2 P CHT Solve ln x = 7 P CHT Solve log x + log(x 15) = 2

29 P CHT Find an exponential growth model for the graph shown y Exponential Growth 9,11,277 0, x P CHT The half-life of radioactive actinium ( 227 Ac) is years. What percent of a present amount of radioactive actinium will remain after 19 years?

30 P CHT Consider an angle that measures 5π 4 radians. (a) Sketch the angle in standard position. (b) Determine two coterminal angles (one positive and one negative). (c) Convert the angle to degree measure. P CHT Find the exact values of the six trigonometric functions of the angle θ shown. y 2,6 Θ x

31 P CHT Given tan θ = 3, find the other five trigonometric functions of θ (Assume θ is 2 in the Quadrant III). P CHT Determine the reference angle θ for the angle θ = 205 and sketch θ and θ in standard position. P CHT Determine the quadrant in which θ lies if sec θ < 0 and tan θ > 0.

32 P CHT Find two exact values of θ in degrees (0 θ 360 ) if cos θ = 3 2. P CHT Find two exact values of θ in radians (0 θ 2π) if csc θ = P CHT Find the remaining five trigonometric functions of θ satisfying cos θ = 3, tan θ < 0. 5 P CHT Find the remaining five trigonometric functions of θ satisfying sec θ = 29, sin θ > 0. 30

33 P CHT Graph (Include two full periods) ( g(x) = 2 sin x π ). 4 Find the amplitude, period, phase shift, and midline. Find the x and y-intercepts and asymptotes (if any). Include a table of nice values for which to evaluate g and the corresponding values of g. Approximate the intervals over which g is increasing, decreasing, or constant. Determine whether the function is even, odd, or neither. P CHT Graph (Include two full periods) f(α) = 1 tan 2α 2 Find the period. Find the α and y-intercepts and asymptotes (if any). Include a table of nice values for which to evaluate f and the corresponding values of f. Approximate the intervals over which f is increasing, decreasing, or constant. Determine whether the function is even, odd, or neither.

34 P CHT Find the exact value of cot ( arcsin 3 8). P CHT Graph ( ) 1 f(x) = 2 arcsin 2 x Find the x and y-intercepts and asymptotes (if any). Include a table of nice values for which to evaluate f and the corresponding values of f. Approximate the intervals over which f is increasing, decreasing, or constant. Determine whether the function is even, odd, or neither.

35 P CHT If tan θ = 6 and cos θ < 0, find all six trigonometric functions of θ. 5 P CHT Simplify P CHT Simlify P CHT Simplify csc 2 β(1 cos 2 β) sec 4 x tan 4 x sec 2 x + tan 2 x cos θ sin θ + sin θ cos θ

36 P CHT Verify P CHT Verify P CHT Simplify sin θ sec θ = tan θ sec 2 x tan 2 x + sec 2 x = sec 4 x csc α + sec α sin α + cos α = cot α + tan α P CHT Verify ( tan x + π ) = cot x 2 P CHT Verify sin(nπ + θ) = ( 1) n sin θ, where n is an integer. P CHT Simplify (sin x + cos x) 2 = 1 + sin 2x

37 P CHT Find all solutions to P CHT Find all solutions to tan 2 x + tan x = 0, sin 2α cos α = 0 on the interval [0, 2π). on the interval [0, 2π). P CHT Find all solutions to P CHT Find all solutions to 4 cos 2 3 = 0, csc 2 x csc x 2 = 0, on the interval [0, 2π). on the interval [0, 2π).

38 P CHT Find the exact value of cos 105 P CHT Use the figure to find the exact values of sin 2u, cos 2u, and tan 2u. y u x 2, 5

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