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College Algebra Problems With Answers
sample 4 : Graphs of Functions

A set of college algebra problems on graphs of functions with answers, are presented. The solutions are at the bottom of the page.
  1. The graph of f(x) is shown below. Draw the graph of y = -f(x - 3) - 2



    college algebra problem 1, function f(x) .






  2. Complete the graph given below so that it is symmetric with respect to the origin.



    college algebra problem 2, function f(x) .






  3. The graph of f(x) is shown below. Sketch the graph of y = f(-x + 1) - 1



    college algebra problem 3, function f(x) .






  4. The graph of f(x) is shown below. Sketch the graph of the inverse of f.



    college algebra problem 4, function f(x) .






  5. The graph of h(x) is shown below.



    college algebra problem 5, function f(x) .



    a) evaluate: h(-2) + h(2) =

    b) What is the domain of h?

    c) What is the range of h?

    d) Find the interval(s) over which h is increasing.

    e) Find the interval(s) over which h is decreasing.

    f) Find the interval(s) over which h is constant.




  6. The graph of f(x) is shown below. Sketch the graph of f(2x).



    college algebra problem 6, function f(x) .






  7. The graph of f -1(x) is shown below.

    Evaluate the following: f(0) , f(2)

    college algebra problem 7, function f(x) .






Answers to the Above Questions
    1. Select points whose coordinates are easy to determine on the given graph (see graph in black below) and then transform them as follows:

      1 - shift right 3 units : f(x - 3)

      2- reflect on the x-axis : - f(x - 3)

      3 - shift down 2 units : - f(x - 3) - 2

      4 - connect the transformed points to sketch the transformed graph shown in red below.

      college algebra problem 1, solution.



  1. A graph is symmetric with respect to the origin if for each point (a,b) on the graph there exists a point (-a,-b) on the same graph.

    We select points (a,b) on the given graph and then transform them into (-a,-b) to obtain more points. When put together the graph is symmeteric with respect to the origin.(see the whole graph black and red below).

      college algebra problem 2, solution.



  2. Select points on the given graph and then transform them as follows:

    1 - reflect on the y - axis : f(-x)

    2- shift right one unit: f(- x + 1) = f(-(x - 1))

    3 - shift down 1 unit : f(- x + 1) - 1

    4 - connect the transformed points to sketch the transformed graph shown in red below.

      college algebra problem 3, solution.



  3. We first determine points (a , b) on the graph of the given function and then use the definition of the inverse to determine points (b , a) on the graph of the inverse. Or use the line y = x to reflect points (a , b) into (b , a).

      college algebra problem 4, solution.



    1. h(-2) + h(2) = -3 + 1 = -2

    2. Domain: [-5 , 5]

    3. Range: {-3} U [0 , 5]

    4. increase: [-3 , 2) , [2 , 5]

    5. decrease: [-5 , -3]

    6. constant: [-2 , 2)



  4. Function f has two x-intercepts: x = 2 and x = -2 . f(2x) will also have x-intercepts such that 2x = 2 which gives x = 1 and 2x = -2 which gives x = -1. Hence f(2x) will have x intercepts at x = 1 and x = -1. The y-intercept is at y = 2 since f(0) = 2 and f(2*0) = f(0) = 2.

      college algebra problem 7, solution.



    1. since f -1(1) = 0 , f(0) = 1 , and since f -1(0) = 2 , f(2) = 0



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Updated: 2 April 2013

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