Section 4-6:  Functions of Two Variables
 
Try the quiz at the bottom of the page!
go to quiz
 
Suppose a company makes two products.  Let x represent the number of the first product made and y represent the number of the second product made.  If the first product makes $2 each and the second $3 each, the profit function would be:
P(x, y) = 2x + 3y
Notice that profit is a function of two variables.  There are a few ways we could graph this function.  One way is to graph in a three dimensional plane.  We will talk about this later in the year.  Another way to graph this function is to hold one of the variables constant.  For example, we could hold y constant like: y = 2, or y = 3 or y = 4, etc.  The equations would look like:
P(x, 2) = 2x + 6
P(x, 3) = 2x + 9
P(x, 4) = 2x + 12
These graphs all have the same slope but different y-intercepts.  These make a family of curves.  They will be graphed in the xP plane.

Another way we can graph this is by holding the profit constant.  For example, we could let P = 6 or P = 12 or P = 18.  Then the equations would be:
6 = 2x + 3y
12 = 2x + 3y
18 = 2x + 3y

 
 Another equation we could use is:
d(r, t) = rt
Suppose we hold t constant at t = 1, t = 2 and t = 3.  Make a graph of these three equations:
d(r, 1) = r
d(r, 2) = 2r
d(r, 3) = 3r

Now graph the same function holding the distance constant at d = 10, d = 20,
and d = 30.  The equations are
10 = rt
20 = rt
30 = rt

 
I'm ready for that practice test now! 
 
I'm almost ready.  I think I will go back and review! 

 
 
Current quizaroo #  4b
 
 
 
 
1)  y + 3 = -f(x - 1)  Describe the translations, symmetry and stretching/shrinking for this graph:
 
a)  Reflected about the y-axis, moved three units up and one unit to the left
b)  Reflected about the y-axis, moved three units down and one unit to the right
c)  Reflected about the x-axis, moved three units up and one unit to the left
d)  Reflected about the x-axis, moved three units down and one unit to the right
e)  Moved three units down and one unit to the right
 
 
 
2) Write the equation of a function that is shifted 2 units up and 4 units to the left, that is reflected about the x-axis with a stretch vertical factor of 2.

          a)  y - 2 = -2f(x + 4)

b)  y + 2 = -2f(x - 4)
c)  y - 2 = 2f(x + 4)
d)  y + 2 = 2f(x - 4)
e)  y +4 = -2f(x - 2)
 

 
 
3)  Find the inverse function to the equation:  y = -3x + 4
a)  y = 3x - 4
b)  y = (4 - x)/3
c)  y = (x - 4)/3
d)  y = (x + 3)/4
e)  y = (3 - x)/4
 
 
 
4)  If g(x) = x2 - 2, x < 0, find the inverse function.
                          ____
a)  y = -\/ x + 2, x > -2
                 ____
b)  y = \/ x + 2, x > -2
 
c)  y = x + 2, x > -2
 
d)  y = -(x + 2), x > -2
 
e)  doesn't have an inverse function, because it doesn't pass the horizontal line test
 
 
  5)  Name an ordered pair that is not in the domain of f(a, b) = (a - b)/(a + b) 
 
a)  (5, 5)
b)  (x, x)
c)  (-d, d)
d)  (-4, -4)
e)  (3/2, 2/3)
 
 
  
 click here for answers!!