University of Maryland Sample Exam 2 Questions, Set 2
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    1. For each of the following functions state whether the function is a polynomial or not. If it is a polynomial, give its degree and leading coefficient. If it is not a polynomial, give a reason why it is not.
      1. f(x) = 31/2 x7 - 10x15 + 17
      2. g(t) =70t + 1
        2t2) + 20
      3. h(z) = 20 (21/2)
    2. Suppose p is a fourth-degree polynomial with
      • x-intercepts -10, 4, 20
      • a negative leading coefficient.
      Draw a possible graph of p and write a function that could have the graph.


    1. Let p(x) = 3x101 - x4 + 4
      1. Describe the behavior of the polynomial's tails.
      2. What is the smallest number of x-intercepts that this polynomial could have?
      3. What is the largest number of turning points that the graph of this polynomial can have?
      4. Find p(-1). Can x + 1 be a factor of p?
      1. Give an example of a rational function that has the asymptote y = 0.
      2. Give an example of a rational function that does not have a horizontal asymptote.


  3. Let
    f(x) =4x2 + 340x + 6000
    1500 + 20x - x2
    1. Find each of the following for the graph of f . Write NONE if the graph does not have that particular intercept or asymptote.
      1. x intercept(s)
      2. y intercept
      3. vertical asymptote(s)
      4. horizontal asymptote
    2. Sketch the graph of f. Show all asymptotes with dotted lines. Label each intercept (for the graph or asymptote) with its value.


    1. Find the exact value of each of the following:
      1. log8 162 p
      2. e31/2 ln 25
      3. log91
        31/7
    2. Find the value of log (5781786) accurate to 4 decimal places. Show clearly how you arrived at your answer.
    3. Suppose log7 c = 5/9. Find the exact values of c and d5.


  5. Use the laws of logarithms to rewrite the following expression in a form with no logarithms of products, quotients or powers:
    log (x2 + 1)4
    103x(x - 3)2


  6. Solve each of the following equations for the exact value of x. If you eliminate a solution, give your reason for doing so.
    1. log2 x + log2(x - 6) = 4.
    2. 18 (1.58x) = 200
    3. 43x-4 = 32x-5


  7. Let h(x) = log5(x +3). Sketch the graph of h and state the domain, range and asymptote of h.


  8. For the functions in parts (a) and (b) find the following information if it applies to the function. If a particular item does not exist for the function state NONE for that item.
    • x-intercepts
    • y-intercept
    • vertical asymptotes
    • horizontal asymptotes
    • table of signs
    • a rough sketch of the graph; be sure to draw the scale on the x-axis carefully and then clearly mark the graph's x-intercepts, its y -intercept, the basic direction of its path as x increases, including where the function is positive and where it is negative and, roughly, where it turns around.
    Do this problem without a calculator; do not depend on your graphing calculator to give you the answers.
    1. f(x) = 0.005(x + 200)(2x - 100)(500 - x)
    2. g(x) = (x - 2)2
      x2 - 5x - 6


    1. Draw the graph of a fifth-degree polynomial f that has
      • x-intercepts -2, 0, 3 and 5,
      • with f(x) > 0 for x < 0
      write a function f that could have the graph.
    2. Let f(x) = log2 ((x - 1)3) - 5.
      1. Find the domain of f
      2. Find two functions g and h (both different from f) such that g(h(x)) = f(x).
      3. Find the x-intercept(s) and y intercept for the graph of f. Label your answer clearly and explain why an intercept does not exist if that is the case.
      4. Use a logarithm property (or properties) to rewrite the formula for f so that the only exponent used is 1. Then explain how the graph of f can be obtained from the graph of k(x) = log2 x.


    1. Find the exact value for each of the following:
      1. log816
      2. ln (1/e)sqrt(2)
    2. Write as a single logarithm and simplify: 3 ln x - (1/2) ln y2 + ln (xy) (for x > 0, y > 0)
    3. If log5 a = 2.34, log5 b = -1.06 use the logarithm laws to find the exact value of each of the following:
      1. log5 (a/25)
      2. log5 (a3/2b)
      3. b1/2


  11. The table below gives the population of the United States from 1800 to 1900.

    year1780179018001810182018301840185018601870188018901900
    Population2.83.95.37.29.612.917.123.231.439.850.262.976.0
      Draw a scatterplot for this data using x = 0 for the year 1800 (that is, plot the points (x, population), where x is the number of years since 1800).
    1. What kind of function will give the best fit to this set of data? Why did you choose this function?
    2. Write an exponential function that goes through the points (0,5.3) and (50, 23.2).
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