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3.2 Max Min

This document provides instruction on determining the maximum and minimum values of quadratic functions. It discusses representing quadratics in standard, vertex, and factored forms and using each form to find the vertex. Methods for finding the vertex by completing the square, factoring, using transformations/partial factoring, and using the quadratic formula are presented. Applications to revenue, cost, and profit functions are also covered. Examples of finding the maximum profit and minimum product of two numbers with a difference of 7 are given.

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Bradley Singh
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129 views2 pages

3.2 Max Min

This document provides instruction on determining the maximum and minimum values of quadratic functions. It discusses representing quadratics in standard, vertex, and factored forms and using each form to find the vertex. Methods for finding the vertex by completing the square, factoring, using transformations/partial factoring, and using the quadratic formula are presented. Applications to revenue, cost, and profit functions are also covered. Examples of finding the maximum profit and minimum product of two numbers with a difference of 7 are given.

Uploaded by

Bradley Singh
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Unit

3 Quadratic Functions







3.2 Determining Maximum and Minimum Values of a Quadratic Function

Date:

Homework: Page 153-154 Questions #3, 4, 5{a,c}, 6, 7{c,d}, 9, 10, 11, 12, 14, 15

Learning Objectives/Success Criteria: At the end of this lesson I will be able to:
o Identify a max/min value of a parabola from standard, factored and vertex form
o Determine the vertex by:
Expressing a quadratic in vertex form by completing the square
Expressing a quadratic in factored form by averaging the roots (zeros)
Partial factoring to determine two points and averaging
Standard Form
Vertex Form
Factored Form
2
2
f (x) = a(x r)(x s), a 0
f (x) = ax + bx + c, a 0
f (x) = a(x h) + k, a 0

The value of the coefficient a indicates the direction of the parabola opens:
Coefficient
Graph
Vertex

a>0



a<0



Determining the vertex of an equation from standard form:
Method 1: Complete the square
1. Factor out the coefficient of the x2 term, or the a
2. Take the coefficient of the x term
Divide by 2
Square it
Add and subtract it
3. Group the perfect square and move the 4th term outside the bracket (multiply it by a)
4. Factor the perfect square and simplify

2
a) f (x) = 3x 2 12x 8





b) g(x) = x 2 + 5x 2
3







Method 2: Factoring and Using the Roots
40
1. Factor the equation
2. Find the midpoint of the roots for the x-coordinate of the vertex
20
3. Sub into the equation to find y-coordinate
-5
5

2
f (x) = 4x 12x 40
- 20

- 40


10

Method 3: Using the Transformations/Partial Factoring


1. Ignore the constant term
2. Factor to determine two points on the parabola

f (x) = 2x 2 12x + 7








" b b 2
%
Method 4: Vertex Formula from Standard Form
$ ,
+ c '
# 2a 4a
&

f (x) = 4x 2 +12x 9

40

20

-5

10

- 20

- 40









Method 5: Using the Quadratic Equation

b b 2 4ac

2a

g(x) = 2x 2 + 8x 5










Applications: Revenue, Cost and Profit Functions
Demand
p(x)
Relation between the price of an item, p, and the number of items sold, x.
Revenue
R(x)
Based on income from sales. Cost times price per item.
Cost
C(x)
Cost to produce a number of items.
Profit
P(x)
Difference between the revenue and cost

Example 1: Nelson publishing company indicates that the demand for their textbooks can be modeled by
p(x) = 5x + 34 and that the cost of producing the textbooks can be modeled by the function C(x) = 3x + 35 ,
where x is the number of textbooks sold in thousands. Determine how many textbooks should be sold to
maximize profits.

Example 2: What is the minimum product of two numbers that have a difference of 7?
What are the two numbers?

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