November 12, 2010
Aim: Optimization Problems

Turning Point
The turning point of a parabola is its vertex.


parabola.png
In this case, the x-coordinate of the vertex lies exactly between the two roots at (3, -1)
You may also use the axis of symmetry equation


Finding Roots Using a Graphing Calculator
Use the left and right buttons to move along the graph and the up and down buttons to toggle between different lines.
1. The 'Zero' Method:
2nd > CALC > 2: zero
Left Bound?: Select a point on the graph below the zero.
Right Bound?: Select a point on the graph above the zero.
Guess?: Select a point between the Left Bound and Right Bound points.
Repeat for other zeroes.
2. The 'Intersect' Method:
Graph y = 0.
2nd > CALC > 5: intersect
First curve?: Select your graph.
Second curve?: Select y = 0
Guess?: Select a point close to their intersection.
Repeat for other intersections.

You can graph a third degree equation and use either method to find its roots.
This is useful for checking Cardano's method questions.

Finding the Turning Points of a Graph

1. Using a graphing calculator
2nd > CALC > 3: minimum
Left Bound?: Select a point to the left of the minimum.
Right Bound?: Select a point to the right of the minimum.
Guess?: Select a point close to the minimum.
2nd > CALC > 4: maximum
Repeat steps to find the maximum of the graph.
2. Using the formula
turningpoint.png

Optimization Problems
1. Number Riddles
Two numbers, x and y, add up to twenty. Which two numbers maximize the product, P?
x + y = 20 ---> y = 20 - x
P = xy
P = x(20 - x)
P = -x^2 + 20x
Restated, what is the x-coordinate of the vertex of parabola P?
Use the equation x = -b/2a
x = -20/-2 = 10
y = 10
P = 100
2. Fencing
The perimeter of a rectangle with dimensions L and W is 80. What is the maximum area, A?
2L+2W = 80 ----> W = 40 - L
A = LW
A = L(40 - L)
A = -L^2 + 40L
Find the vertex of parabola A
L = -40/-2 = 20
W = 20
A = 400

Wendi Liu
Period 10