1. Problem Solving and Quadratic Functions
Section 11.8
Problem Solving and Quadratic Functions

2. Maximum and Minimum Problems
Maximum Problems
A maximum value exists if a of the equation ax²+bx+c=0 is negative.
The maximum value is defined as the vertex.
Minimum Problems
A minimum value exists if a of the equation ax²+bx+c=0 is positive.
The minimum value is defined as the vertex.

3. Example 1
Sweet Harmony Crafts has determined that when x hundred dulcimers are built, the average cost per dulcimer can be estimated by C(x) = 0.1x² – 0.7x , where C(x) is in hundreds of dollars.
What is the minimum average cost per dulcimer
How many dulcimers should be built in order to achieve that minimum?

4. Example 1 What is the minimum average cost per dulcimer
Find the y part of the vertex (h , k)
C(x) = 0.1x² – 0.7x
h = -b / 2a = / 2(0.1) = 0.7/0.2 = 3.5
k = C(3.5) = 0.1(3.5)² – 0.7(3.5)
= 0.1 (12.25) – 0.7(3.5)
= – = 1.2
The minimum average cost is in hundreds of dollars (1.2), $120 per dulcimer.

5. Example 1
How many dulcimers should be built in order to achieve that minimum?
Find the x part of the vertex
h = -b / 2a = / 2(0.1) = 0.7/0.2 = 3.5
3.5 hundred dulcimers should be built in order to achieve that minimum.
350 dulcimers should be built in order to achieve that minimum.

6. Example 2
What is the maximum product of two numbers that add to 26? What numbers yield this product?

7. Example 2 Product of two numbers xy Add up of two numbers x + y
Equation x + y = 26
Product xy
Isolate s or y in the equation and substitute into the product, solve.

8. Example 2 Equation x + y = 26 x = 26 - y Product xy (26 - y)y
26y - y² Solve for x of vertex
- y² + 26y + 0
-b/2a = -26 / 2(-1) = 13
The two maximum numbers are 13 and 13

9. Homework
Section 11.8
7, 8, 17, 19, 23, 24, 25,