University of Calgary

Continuing Education

Math 2

Assignment 1

Transformations

1. Consider the graph of the function y = f(x)

1. 1. Describe the transformation of y = f(x)   to  ?              =              3?(−2(?           −           1))        +           4 .

1. 2. Sketch the new graph

2. If the range of the function y = f(x) is {y | y ≥ 4 }, state the range of the new function        

                   ?(?)       =   ?(?   +     2)      −           3.

3. Consider ?(?) =           ?⁰          −           9 .
   1. Sketch the graph of  f(x).

1. 2. Determine the equation of the inverse of f(x) and sketch its graph.

1. 3. State the equation of ?            =           2?(?) and sketch its graph.

1. 4. Identify and compare the domain and range of the three relations.

4. As a result of the transformation of the graph of y = f(x) into the graph of                                ?            =           −3?(? +           2)              −           5 , the point (2, 5) becomes the point (x, y).                  Determine the value of (x, y).

Functions

1. Given ?(?)  =           2?⁰ +   11?       −           21 and ?(?)    =              2?         −           3, determine the equation, state  the domain and range and sketch the graph of each combined function.  a) ?              =           ?(?)     −           ?(?)

              b)          ?            =           ?(?)     +           ?(?)     c)           ?            =             

               d)           ?  =  ?(?(?))

2.          Given ?

  and ?(?)  =           4            −           ? , match the combined functions in Set A with              the graph in Set B.

               Set A

              i)            (?          −           ?)         (?)                                                  ii)               (?          +           ?)         (?)                      iii)          ?(?)?(?)                                                     iv)         

          (?) 

Polynomial Functions

1. The partial graph of a third degree polynomial function

               ?(?)    =           ?(? − ?)(? − ?)(? − ?) is shown.  Determine the value of a, and write the                polynomial in factored form. 

2. Given the polynomial function ?(?)  =           ?⁼ + ?⁰ −              10?       +           8, determine the x and y                intercepts, and the zeros of the function.  Determine the intervals where the function is     positive and the intervals where the function is negative.

3. Given that ?     +           3 is a factor of the polynomial ?(?)   =              ?^@ + 3?⁼          +           ??⁰ −    7?         +           6 ,  determine the value of c.  Then, factor the polynomial fully.

4. When ?(?) =  ?⁼         −           7?⁰       +           ??         +           17 is divided by ? −           5, the remainder is 2.   Determine the value of k.

Exponential and Logarithmic functions

1.        Match each of the following equations with its graph.

           a)        ? = 5(2⁵) + 1                              b) ? = _D^E_0F^5GH

           c)        ? + 1 = 2^HI5                                d) ? = 5 _D0^E_F^I5

2. Express in logarithmic form

1. 1. ?            =           3⁵                                                    b)          ?          =              2^KGE

3. Express in exponential form

1. 1. log₅ 3 =             4                                                      b)          log_K(? + 5) =     ?

4. Express each expression as a single logarithm in simplest form.  

1. 1. 2 log ? − (log ? + 3 log ?)    

               b)           2 log(? + 1) + log(? − 1) − log(?⁰ − 1)

5. The diagram shows part of the graph of ?      =           log_Q(? − ?) where ?, ??              ?

          When the original function was ?       =           log_Q ? , use                the points on the graph and your knowledge

              of transformations to determine the value of             a + 3b.

6. Solve each equation for the value of x

               a)           log₅ 16 = 4                                               b)                  7^5GE =            4^05IE

               c)            4^05GE =    9(4^EI5)                                      d)   2 log_H(? − 3) = log_H(4)

               e)           log₀(? − 4) =  4               −        log₀(? + 2)  

               f)                        log_@(? + 2) −      log_@(? − 4)              =           ^E₀

7. The estimated number of bacteria present in a certain culture after t hours is given by

              ?(?)     =           ?(?^)?Z?, where C is the initial number of bacteria, t is the time and d is the      doubling period.  A particular culture of 100 has a doubling period of 2 hr.  Find the time                required to reach 30 000.  Express your answer to the nearest tenth of an hour.