PreCalculus Class Notes QF3 Writing and Solving Quadratic Models
Using the vertex of a parabola, you can answer questions about minimum and maximum values.
This is called optimization.
minimum if a is positive (a > 0)
maximum if a is negative (a < 0)
Maximizing Area for a Given Perimeter
Applets: http://mste.illinois.edu/users/carvell/rectperim/RectPerim.html and/or GSP
RiverPenArea
Example
A rancher is fencing a rectangular area for cattle using the straight portion of
a river as one side of the rectangle. If the rancher has 2400 feet of fence, find
the dimensions of the rectangle that give the maximum area for the cattle.
Solution
Draw a diagram and label L and W. Write an equation for perimeter and an equation for area.
Use information about the vertex to find the dimensions for the maximum area.
Projectile Motion
General Formula for Path of an Object with Gravity
s ( t ) = −16t 2 + v0t + s0
where
s(t) is the height (feet) of the object at time, t (seconds).
v0 is the initial vertical velocity (ft per sec).
Positive velocity if object is thrown up and negative velocity if the object is thrown down.
s0 is the initial height (feet) of the object.
Example
A baseball is hit straight up with an initial velocity of 80 feet per second (or about 55 miles per
hour) and leaves the bat with an initial height of 3 feet.
a) Write a formula s(t) that models the height of the baseball after t
seconds.
s ( t ) = −16t 2 + v0t + s0
b) How high is the baseball after 2 seconds?
c) Find the maximum height of the baseball. Support your answer graphically.
Time for CBR bouncing the ball? s ( t ) = −16t 2 + v0t + s0