Problem Solving and Quadratic Functions Section 11.8 Problem Solving and Quadratic Functions

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.

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 + 2.425, 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?

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 + 2.425 h = -b / 2a = --0.7/ 2(0.1) = 0.7/0.2 = 3.5 k = C(3.5) = 0.1(3.5)² – 0.7(3.5) + 2.425 = 0.1 (12.25) – 0.7(3.5) + 2.425 = 1.225 – 2.45 + 2.425 = 1.2 The minimum average cost is in hundreds of dollars (1.2), $120 per dulcimer.

Example 1 How many dulcimers should be built in order to achieve that minimum? Find the x part of the vertex h = -b / 2a = --0.7/ 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.

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

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.

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

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