MATH 0 OLD EXAM QUESTIONS FOR EXAM 3 ON CHAPTERS 3 AND 4 PAGE Questions From Old Eams. Write the equation of a quadratic function whose graph has the following characteristics: It opens down; it is stretched by a factor of ; its ais of symmetry is the line = ; it passes through the point (4, -).. Epress area A of an equilateral triangle as a function of where is length of one side of triangle. 3. Find the Average Rate of Change of 4. Given the function f f from to 3. 3 a. At what point does the graph cross the -ais? b. At what point does the function touch the -ais? c. What is the behavior of the function near each of its zeros? That is, what function does f look like near each zero?. Given the function f 0: a. Write the function in verte form. b. What are the coordinates of the verte? c. What is the equation of the ais of symmetry. d. What are the coordinates of the -intercepts? e. What are the coordinates of the y-intercept? 3 3 f : 3 3 9 33 a. What are the equations of the vertical asymptotes? b. What are the coordinates of the -intercepts? c. What are the coordinates of holes in the graph, if any? d. What is the equation of the horizontal or oblique asymptote? 3 6. On your graph, be sure to clearly show asymptotes, holes, intercepts, and indicate where the graph crosses an asymptote if it does. 6. Given the function 7. Sketch the graph of the function f 8. Consider the graph of a rational function that has the following characteristics. Show each of the following on the grid below. Be sure each characteristic is accurately shown on the graph. Hole at, 0 Crosses the -ais at, 0 Vertical asymptote 0 with even multiplicity Horizontal asymptote: y 0 Graph crosses horizontal asymptote at, 0 Sketch the graph: Write a function that could produce such a graph. 9. Solve: Write your answer using interval notation.
MATH 0 OLD EXAM QUESTIONS FOR EXAM 3 ON CHAPTERS 3 AND 4 PAGE Solutions. Write the equation of a quadratic function whose graph has the following characteristics: It opens down; it is stretched by a factor of ; it s ais of symmetry is the line = ; it passes through the point (4, -). f ah k y k When 4, y 4 k 6 k f 6. Epress the area A of an equilateral triangle as a function of where is the length of one side of the triangle. A bh h 0-0 - 0 To find the height, use Pythagoras: c a b h 4 3 h 4 3 h h 3 3 A bh h 4 3. Find the Average Rate of Change of f from to 3. 3 The ARC of a linear function is the slope of the line. In this case, m. 3
MATH 0 OLD EXAM QUESTIONS FOR EXAM 3 ON CHAPTERS 3 AND 4 PAGE 3 4. Given the function f a. At what point does the graph cross the -ais? The zero at = - has odd multiplicity so the graph crosses the -ais at (-, 0). b. At what point does the function touch the -ais? The zero at = has even multiplicity so the graph touches the -ais at (, 0). c. What is the behavior of the function near each of its zeros? f look like That is, what function does near each zero? Near = - the function looks like - 9 This is a very steep reflected cubic shape. f. Near = the function looks like 43 This is a very steep upside down parabolic shape. f. 0-0 - 0 - -0
MATH 0 OLD EXAM QUESTIONS FOR EXAM 3 ON CHAPTERS 3 AND 4 PAGE 4. Given the function f 0: a. Write the function in verte form. f 0 6 0 6 99 0 6 9 9 0 3 8 b. What are the coordinates of the verte? We can read the verte directly from the equation: 3,8. Or, the -coordinate is given by b verte 3 a To find the y-coordinate, calculate f verte f 33 30 8 36 0 8 c. What is the equation of the ais of symmetry. The ais of symmetry is the vertical line that passes through the verte. The equation of any vertical line is = k, where k is a constant. In this case k is the value of verte. So, the equation of the ais of symmetry is = 3. d. What are the coordinates of the -intercepts?,0,,0 The -intercepts are the values of when y = 0. 0 0 0 6 0 or e. What are the coordinates of the y-intercept? The y-intercept is the value of y when = 0: f 0 y 0 0 0 y 0
MATH 0 OLD EXAM QUESTIONS FOR EXAM 3 ON CHAPTERS 3 AND 4 PAGE 6. Given the function f 3 3 : 3 3 9 3 3 a. What are the equations of the vertical asymptotes? 3 The function reduces to g. The denominator is 0 when = - and = -3. 3 3 So, the vertical asymptotes are the lines = - and = -3. b. What are the coordinates of the -intercepts? These are the zeros of the numerator. They are (3, 0) and (-, 0). c. What are the coordinates of any holes in the graph, if any? The common factor of cancels so there is a hole when = 0. The y-coordinate is g 0. g 03 0 3 4 0 3 0 0 3 3 3 3. So, the hole is at 4 0, 3. d. What is the equation of the horizontal or oblique asymptote? The degree of the numerator and denominator is the same so there is a horizontal asymptote whose value is the ratio of the coefficients of the dominant terms. This is the line y. 3 0-0 - 0 - -0
MATH 0 OLD EXAM QUESTIONS FOR EXAM 3 ON CHAPTERS 3 AND 4 PAGE 6 7. Sketch the graph of the function f 3 6. Step. Factor, state domain, and THEN reduce. f g 6 6 3 6 6 There is a hole when - = 0, which means when =. The y-coordinate of the hole is g 6 7 So, the hole is at 7,. Step. Plot intercepts. For -intercepts, state whether graph touches (multiplicity is even) or crosses (multiplicity is odd) -ais. -intercepts at = 0 and = -6. Both have multiplicity odd so graph crosses at (0, 0) and (-6, 0). Step 3. Draw vertical asymptotes (the zeros of the denominator). The denominator of the reduced fraction is 0 when = - so = - is a VA. Step 4. Draw horizontal asymptotes: deg num = deg denom +, so use long division to find oblique asymptote, y = m + b 6 6 6 g 6 So, the OA is y = - -. Step. Plot where the graph crosses an HA or OA.. g y 6 6 6 0 This has no solution so the graph does not cross the OA. Step 6. If needed, plot a few etra points. Step 7. Connect the dots. 0 - -0-0 - -0 -
MATH 0 OLD EXAM QUESTIONS FOR EXAM 3 ON CHAPTERS 3 AND 4 PAGE 7 8. Sketch the graph on the grid below. There are two possible solutions: y 0 0 - Cross 0 - -0 - Write a function that could produce such a graph: Ans: R Hole at = implies ( - ) in both denominator and numerator. R or Crossing the -ais at, 0 implies other factor of - in numerator. R Vertical asymptote 0 implies in denominator. Even multiplicity suggests second power: R Horizontal asymptote: y 0 says degree of numerator is less than the degree of denominator. This checks with R Graph crosses the horizontal asymptote at, 0. This checks: Function Horizontal Asymptote y R 0 0 - -
MATH 0 OLD EXAM QUESTIONS FOR EXAM 3 ON CHAPTERS 3 AND 4 PAGE 8 9. Solve: Write your answer using interval notation. Step. Write inequality with 0 on one side. 0 0 0 0 0 Step. Find the real zeros of the numerator and denominator. = - and = 0 Step 3. Use the real zeros to break up the number line into intervals. Interval : 0 Step 4. Use a test point in each interval to see if the function is pos or neg. Interval : 0 Try / // Get f / Simplify : 4 f 0 yes yes no We want 0 f so,,0-0 - 0-0. Note that f(-) = 0 so we must eclude -. -