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1 MAC 1147 Exam #1a Answer Key Name: Answer Key ID# Summer 2012 HONOR CODE: On my honor, I have neither given nor received any aid on this examination. Signature: Instructions: Do all scratch work on the test itself. Make sure your final answers are clearly labelled. Be sure to simplify all answers whenever possible. SHOW ALL WORK ON THIS EXAM IN ORDER TO RECEIVE FULL CREDIT!!! No. Score 1 /4 2 /6 3 /8 4 /2 5 /2 6 /3 7 /2 8 /2 9 /2 10 /2 11 /2 12 /2 13 /6 14 /14 15 /8 16 /8 17 /25 18 /2 19 /4 20 /8 21 /2 22 /2 23 /2 24 /10 25 /4 26 /4 27 /2 28 /2 29 /2 30 /8 Total /100

2 (1) Determine whether the relation represents a function. If it is a function, state the domain and range. (2 points each) (i) Determine whether the relation represents a function. If it is a function, state the domain and range. {( 4, 2), ( 1, 1), (0, 0), ( 1, 1), ( 4, 2)} (a) not a function (b) function domain: { 4, 1, 0} range: {2, 1, 0, 1, 2} (c) function domain: {2, 1, 0, 1, 2} range: { 4, 1, 0} (ii) Determine whether the equation x + y = 6 defines y as a function of x. (a) No (b) Yes

3 (2) Find the value of the function. (2 points each) (i) Find f( 1) when f(x) = x2 4 x 3. (a) 1 4 (b) 5 4 (c) 3 4 (d) 5 2 (ii) Find f( x) when f(x) = 2x 2 3x + 4. (a) 2x 2 + 3x + 4 (b) 2x 2 + 3x 4 (c) 2x 2 + 3x 4 (d) 2x 2 + 3x + 4 (iii) Find f(x 1) when f(x) = 5x 2 3x + 4. (a) 5x 2 13x + 12 (b) 5x x + 6 (c) 13x 2 + 5x + 12 (d) 5x 2 13x + 6

4 (3) Find the domain of the function. (2 points each) (i) f(x) = 4x 2 16 (a) [2, ) (b) (2, ) (c) (, 2) (d) (, 2) ( 2, 2) (2, ) (e) (, ) (ii) f(x) = 3x x 2 +9 (a) (, 3) ( 3, 3) (3, ) (b) (, 3] [ 3, 3] [3, ) (c) (, 3) (3, ) (d) x = 3, 3 (e) (, ) (iii) f(x) = 25 5x (a) (, ) (b) (, 5) (c) (, 5] (d) (5, ) (e) [5, ) (iv) f(x) = 9 x 3 (a) (, ) (b) (3, ) (c) [3, ) (d) (, 3) (e) (, 3]

5 Use the following graph to answer questions (4) Does this graph represent a function? (2 points) (a) Yes (b) No (5) What is the domain and range of the graph? (2 points) (a) domain: (, 4) range: ( 6, 6) (c) domain: (, 4] range: [ 6, 6] (b) domain: ( 6, 6) range: (, 4) (d) domain: [ 6, 6] range: (, 4] Domain = (, 4]; Range = (, 6] (6) What are the x- and y-intercepts of the graph? Note: Not multiple choice. (3 points) x-intercepts: ( 4, 0), ( 2, 0), (0, 0), (2, 0), (4, 0) y-intercept: (0, 0)

6 (7) What type of symmetry does the graph have? (2 points) (a) x-axis (b) even function (c) odd function (d) no symmetry (8) What is the value of f(1)? (2 points) (a) 0 (b) 1 (c) 6 (d) none of the above (9) Is f ( 7) greater than 2 or less than 2? (2 points) (a) f ( 7) > 2 (b) f ( 7) < 2 (10) For what numbers x is f(x) > 0? (2 points) (a) [ 4, 2] [4, 5] (b) (, 4] (c) (, 4] [ 2, 0] [2, 4] (d) ( 4, 2) (0, 2) (e) (, 4) ( 2, 0) (2, 4) (11) For what numbers x is f(x) 0? (2 points) (a) [ 4, 2] [4, 5] (b) (, 4] (c) (, 4] [ 2, 0] [2, 4] (d) ( 4, 2) (0, 2) (e) (, 4) ( 2, 0) (2, 4) (12) On what intervals is the function increasing and decreasing? (2 points) (a) increasing: (, ) ( 1, 6) (3, 4) decreasing: (4, 6) (6, 4) (b) increasing: (4, 6) (6, 4) decreasing: (, ) ( 1, 6) (3, 4) (c) increasing: (, 3) ( 1, 1) (3, 4) decreasing: ( 3, 1) (1, 3) (d) increasing: ( 3, 1) (1, 3) decreasing: (, 3) ( 1, 1) (3, 4) (13) State whether each point is a local maximum, local minimum, or neither. (1 point each) (i) ( 4, 0)

7 (a) local maximum (b) local minimum (c) neither (ii) ( 3, 4) (a) local maximum (b) local minimum (c) neither (iii) ( 1, 6) (a) local maximum (b) local minimum (c) neither (iv) (1, 6) (a) local maximum (b) local minimum (c) neither (v) (3, 4) (a) local maximum (b) local minimum (c) neither (vi) (4, 0) (a) local maximum (b) local minimum (c) neither

8 (14) Let f(x) be the piecewise-defined function { x 3 1 if 2 x < 0 f(x) = x if 0 x 4 (i) Find f(4). (2 points) 2 (ii) Sketch the graph of f(x). (8 points) (iii) State the domain and range of f(x). (4 points) Domain = [ 2, 4]; Range = ( 1, 7]

9 (15) Use the graph of the function f illustrated below to graph the function H(x) = 2f( x) 3. (8 points)

10 (16) For the function below, state what types of transformations have been done to the parent function to arrive at the given function. Then graph the function. Be sure to state what order the transformations are done in. (8 points) f(x) = 1 2 x 3 1 st : horizontal stretch 2 nd : reflection about x-axis 3 rd : down 3

11 (17) Let f(x) be the quadratic function given below. Answer the following questions about f(x). f(x) = 2x 2 4x + 6 (i) Does the graph of f open up or down? How do you know? (2 points) Down (ii) What is the vertex (h, k) of f? (4 points) ( 1, 8) (iii) What are the intercepts of f? (4 points) x-intercepts: ( 3, 0), (1, 0); y-intercept: (0, 6) (iv) What is the domain of f? (1 point) (, ) (v) What is the range of f? (2 point) (, 8]

12 (vi) What are the intervals of increase and decrease of f? (4 points) Increasing: (, 1); Decreasing: ( 1, ) (vii) What does the graph of f look like? (8 points)

13 (18) For the polynomial function given below, list each real zero and its multiplicity. Determine whether the graph crosses through the x-axis or touches the x-axis and turns around at each x-intercept. (2 points) f(x) = 4(x + 7)(x 1) 4 (a) 7, multiplicity 1, touches the x-axis and turns around; 1, multiplicity 4, crosses through the x-axis (b) 7, multiplicity 1, crosses through the x-axis; 1, multiplicity 4, touches the x-axis and turns (c) 7, multiplicity 1, touches the x-axis and turns around; 1 multiplicity 4, crosses through the x-axis (d) 7, multiplicity 1, crosses through the x-axis; 1, multiplicity 4, touches the x-axis and turns around (e) None of the above

14 (19) For parts (i) and (ii), let f(x) = x 2 (x + 3) 3 (x 2 1) (i) For the polynomial function given above, determine the end behavior; that is, find the power function that the graph of f resembles for large values of x. (2 points) (a) y = x 3 (b) y = x 7 (c) y = x 3 (d) y = x 7 (ii) Determine the graph of the power function in part (a). (2 points) (a) (b) (c) (d)

15 (20) Graph the polynomial function given below. (8 points) f(x) = (x 2)(x + 2) 2 (x 1) 2

16 (21) Find the vertical asymptotes, if any, of the rational function. (2 points) f(x) = 16 x2 x 2 + 5x + 4 (a) x = 1 (b) x = 1, x = 4 (c) x = 1, x = 4 (d) x = 1, x = 4 (e) None of the above (22) Find the horizontal or oblique asymptote, if any, of the rational function. (2 points) 3x 2 f(x) = x 2 + 4x 45 (a) y = 3x + 1 (b) y = x 3 (c) y = 0 (d) y = 3 (e) None of the above (23) Find the horizontal or oblique asymptote, if any, of the rational function. (2 points) f(x) = 2x3 + 11x 2 + 5x 1 x 2 + 6x + 5 (a) y = 0 (b) y = 2 (c) y = 2x 1 (d) y = 2x + 1 (e) None of the above

17 (24) Graph the function. (Hint: You will need to determine the domain, vertical asymptotes and/or holes, intercepts, and end behavior.) (10 points) f(x) = 3x2 (x 1)(x + 4) 3 (x + 2) (x + 1) 4 (x 2) 2 (x + 2)

18 (25) Solve each inequality algebraically. (2 points each) (i) x 3 6x 2 (a) (0, 6] (b) (, 0] [6, ) (c) (, 6] (d) [6, ) (e) None of the above (ii) (x + 2) 2 (x + 4) < 0 (a) (, 4) (b) ( 2, ) (c) (, 4) ( 4, 2) (d) ( 4, 2) ( 2, ) (e) None of the above

19 (26) Solve each inequality algebraically. (2 points each) (i) x+9 x+5 < 2 (a) (, 5) ( 1, )(b) ( 5, 1) (c) (, 5) (5, 8) (d) (, 1) (5, ) (e) None of the above (ii) x2 (x 12)(x+3) (x 4)(x+8) 0 (a) ( 8, 3] (4, 12] (b) (, 8) [ 3, 4) [12, ) (c) (, 8] [ 3, 4] [12, ) (d) (, 8) [ 3, 0) (0, 4) [12, ) (e) None of the above

20 (27) Give the maximum number of zeros the polynomial function may have. Use Descartes Rule of Signs to determine how many positive and how many negative zeros it may have. (2 points) f(x) = x 6 + 4x 5 + 4x 4 + 3x 3 x 2 5x + 4 (a) 6; 2 or 0 positive zeros; 5, 3, or 1 negative zeros (b) 6; 3 or 1 positive zeros; 4, 2, or 0 negative zeros (c) 6; 4, 2, or 0 positive zeros; 2 or 0 negative zero (d) 6; 2 or 0 positive zeros; 4, 2, or 0 negative zero (e) None of the above

21 (28) List the potential rational zeros of the polynomial function. (2 points) f(x) = 2x 3 + 3x 2 2x + 8 (a) ± 1, ± 1, ±1, ±2, ±4, ±8 (b) ± 1, ±1, ±2, ± (d) ± 1, ± 1, ± 1, ±1, ±2, ±4, ±8 (e) ± 1, ±1, ±2, ±4, ± (29) Use the given zero to find the remaining zeros of the function. (2 points) f(x) = x 4 45x 2 196; zero: 2i (a) 2i, 14, 14 (b) 2i, 7, 7 (c) 2i, 7i, 7i (d) 2i, 14i, 14i (e) None of the above

22 (30) Find the real and complex zeros of the polynomial function and write f in factored form. (8 points) f(x) = x 4 x 3 x 2 x 2 zeros: x = 2, 1, i, i f(x) = (x 2)(x + 1)(x i)(x + i)

23 Extra Blank Graphs.