1. Quadratic Functions – Maximum and Minimum Word Problems
1. Projectile Motion
 Maximum Height
2. Numbers
 Maximizing or Minimizing number combinations
3. Maximizing Area
4. Maximizing Revenue

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3. Day 1 There are 4 types of maximum/minimum word problems:
1. Projectile Motion
 Maximum Height
2. Numbers
 Maximizing or Minimizing number combinations
3. Maximizing Area
4. Maximizing Revenue

4. Steps to solving Maximum and Minimum Word Problems
Identify the quantity to be maximized or minimized and write an algebraic expression for this quantity.
Rewrite the expression using only 1 variable (ie. "x”)
………………
Expand and simplify to change the second equation into standard form. y = ax2 + bx + c
Complete the square to maximize / minimize the function. y = a(x – h)2 + k
Make a maximum / minimum statement to fully answer the question.

5. Type 1: Projectiles
Example 1: A football is kicked so that its height, h meters after t seconds is given by the expression: h = 22t - 5t2.
a) What is the height of the Football when it is kicked?
When the football is kicked, t = ………………
h = 22t - 5t2
h = 22(0) - 5(0)²
h = 0
The height of the football when it is kicked ………………

6. b) What is the maximum height of the football?
h = 22t - 5t2
h = - 5t2 + 22t
………………
h = -5[t t – 4.84]
h = -5(t - 2.2)
The maximum height is 24.2 m above the ground

7. c) When does the football reach its maximum height? h = 24.2
………………
The ball reaches its maximum height 2.2 seconds after it is kicked

8. d) Sketch the graph.
The vertex of the football’s path is (2.2, 24.2)
One x-intercept is (0, 0) because that’s time = 0
Since the axis of symmetry is x = 2.2, the other x-intercept is = 4.4
or set h = 0
0 = -5(t - 2.2)
-24.2/-5 = (t-2.2)2
………………

9. e) How long is the ball in the air?
Since graph is symmetrical about its axis of symmetry, x= 2.2, and the ball is on the ground when t = 0, it will hit the ground again when ………………
Or think of it like this, it will take an object the same amount of time to go up as come down.

10. Type 2: Numbers
Example 1: The sum of two numbers is twelve. Find the numbers and the maximum of their product.
Step 1. Assign the variables
………………
Let y represent the second number.
Step 2. Write the info in terms of only one variable (x). Since the sum of the two numbers is twelve:
x + y = 12
12 – x = y
y = 12- x

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