2. The sum of two or more functions can be
found by adding the ordinates of the
functions at each abscissa.....
In English this means...
To find the sum (or new function) of a
minimum of two other functions you have to
add the y intercepts of each function at each
and every x intercept.
3. Example 1.a)
The equation h(x) can be found by adding the expressions for f(x) and
g(x).
One of the best ways to see how these functions
combine to create the function h(X) is to graph the
two original functions on the same axis.....
4. Once you have the graph of the original functions, you
can begin to find the new equation.
When you begin working with adding functions one of
the best ways to do this is to add them using a table of
values.
The next page will have
graphed on the same axis and a table of values
beginning to determine the graph of h(x).
5. This is the of f(x) graph
The below table shows the value
of each equation at a certain x
intercept.
This is the graph of g(x) x5 f(x)=x2 g(x)=3 h(x)=x2+3
-3 9 3 9+3=12
On the next slide
All three of the -2 4 3 4+3=7
expressions will be -1 1 3 1+3=4
graphed on the
0 0 3 0+3=3
same axis.
1 1 3 1+3=4
2 4 3 4+3=7
3 9 3 9+3=12
6. Although it is not overly clear from
this graph, this superposition can
be thought of as a constant vertical
translation of the parabola y=x2 to
produce the new parabola.
What this means, is that each point
on the y=x2 parabola has been
translated up 3.
x f(x)=x2 g(x)=3 h(x)=x2+3
-3 9 3 9+3=12
-2 4 3 4+3=7
-1 1 3 1+3=4
0 0 3 0+3=3
1 1 3 1+3=4
2 4 3 4+3=7
3 9 3 9+3=12
7. The superposition principle
can be used to find the
difference of two
functions, because
subtracting is the same as
adding the opposite.
8. Mr. Mitchell is selling Zombie shirts and survival packs to raise
money for the math department. The fixed cost of the shirts and
packs is $200 and there is an additional variable cost of $5 per item
made. Mr. Mitchell decides to sell the Zombie paraphernalia for $8
each.
a) Write an equation to represent... And then graph the function on the same
axis. Label and explain what the point of intersection represents.
- Total cost, C, as a function of the number, n, of items produced
- Revenue, R, as a function of the number, n, of items produced
b) Profit, P, is the difference between revenue and expense. Develop and
algebraic and graphical model for the profit function. (Solve and graph for
profit).
c) What sales would result in a loss for Mr. Mitchell and math? A profit?
9. a)
The total cost of producing the Zombie items is the sum of the fixed
cost and the variable cost: C(n)=200+5n
The revenue is $8 per item, multiplied by the total number of items
sold: R(n)=8n
Use a graphing calculator to graph these two functions, use
appropriate window settings.
C(n)=200+5n
R(n)=8n
10. Determine the point of intersection algebraically
To find the POI sole the linear system, the expression will be the same when R=C
8n= 200+ 5n
3n= 200 C(n)= 200+5n
n= 200 R(n)= 8n
3
R= 8n
= 8(200/3)
= 1600 Find the corresponding value of R.
3
C(n)=200+5n
The POI is 200 , 1600
3 3
These coordinates show the point at which a profit will
begin to be made.
R(n)=8n
11. b) Profit is the difference between revenue and expenses
P(n)=R(n)-C(n) You can use the superposition to
=8n – (200+5n) model this new function
=8n-200-5n graphically.
=3n-200
This new function shows the profit being made.
12. c) Mr. Mitchell and the math department will lose money when the
P(n) graph is below the x-axis, or when 67 or less items have been
sold. The math department will start making a profit when more
than 67 items have been sold.
In summary, if the sale of Zombie paraphernalia is to be successful
then the math department needs to sell at least 68 items.
13. Key Concepts
All of the work we just did can be
found on pages 418-423 in our
textbook.
