Chapter 3: Functions and Graphs

 

1.      Plot the points A(-6, 2), B(6,2), C(-4, -6), D(0, 6), and E(4, -6).

 

2.      Find the point with coordinates of the form (2a, a) that is in the third quadrant and is a distance 5 from P(1, 3).

 

3.  Sketch the graph of the equation:

     a) y = 2x - 3        b) y = -x 3 + 1        c) y = √x    

 

4.      Sketch the graph of the circle:

a) x 2 + y 2 = 9                            b) 5x 2 + 5y 2 = 25

 

5.      Find an equation of the circle that satisfies the given conditions:

a)     Center C(2, -3), radius 5.

b)     Center at the origin, passing through P(8, -14).

 

6.      Find the center and radius of the circle with the given conditions:

a)     x 2 + y 2 - 4x + 6y - 36 = 0

b)     x 2 + y 2 + 4x + 6y + 16 = 0

 

7.      Find the equations for upper half, lower half, right half, and left half of the circle:

x 2 + y 2 = 36

 

8.      Find x-intercept and y-intercept for x 2 + y 2 - 4x - 6y + 4 = 0

 

9.  Find an equation of a line through A(1, 7) and B(-3, 2).

 

10.  Find an equation of a line through A(7, -3) and perpendicular to the line

       2x - 5y = 8.

 

11.  Find the domain of f(x) = sqrt(2x + 7).                    *sqrt = square root of

 

12.  (a) Sketch the graph. (b) Find the domain and range. (c) Find the intervals on   

       which it is increasing, decreasing, or constant.  f(x) = 3x - 2.

 

13.  (a) Sketch the graph. (b) Find the domain and range. (c) Find the intervals on  

       which it is increasing, decreasing, or constant.  f(x) = sqrt(x + 4).

 

14.  Sketch the graph of:

 a) f(x) = x 2     b) f(x) = x 2 + 4     c) f(x) = x 2 - 4    

 d) f(x) = x 3     e) f(x) = x 3 + 4      f) f(x) = x 3 - 4

 

 

 

15.  A doorway has the shape of a parabolic arch and is 9 feet high at the center   

      and 6 feet wide at the base.  If a rectangular box 8 feet high must fit through     

       the doorway, what is the maximum width the box can have?

 

16.           __ x    1   2   3   4 _                ___x    1    2   3   4_

                   f(x)   4   3   2   1                     g(x)    3   4   2   1

       Find (f ◦ g)(3), (g ◦ f)(3), (f ◦ f)(3), and (g ◦ g)(3).

 

16.  Prove that the function f(x) = 2x - 3 is one-to-one.

 

17. Find the inverse function of f(x) = 2x - 3.

 

18.  Express the statement as a formula:  y is directly proportional to the square   

       root of x and inversely proportional to the cube of z.  If x = 9 and z = 2, then

       y = 10.

 

19.  Express the statement as a formula:  r is directly proportional to the product   

       of s and v and inversely proportional to the cube of p.  If s = 2, v = 3, and

       p = 5, then r = 40.

 

20.  A circular cylinder is sometimes used in physiology as a simple  

       representation of a human limb. 

a)     Express the volume V of a cylinder in terms of its length L and its circumference C.

b)     The formula obtained in part (a) can be used to approximate the volume of a limb from length and circumference measurements.  Suppose the (average) circumference of a human forearm is 22 centimeters and average length is 27 centimeters.  Approximate the volume of the forearm.