2.3 Quadratic Functions by Factoring

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2.3 Quadratic Functions by Factoring
ACT
2
Objectives

• F-IF.4 For a function that models a relationship
  between two quantities, interpret key features of
  graphs and tables in terms of the quantities, and
  sketch graphs showing key features given a verbal
  description of the relationship. Key features
  include intercepts intervals where the function
  is increasing, decreasing, positive, or negative
  relative maximums and minimums symmetries end
  behavior and periodicity.
• A-CED.1 Create equations and inequalities in one
  variable and use them to solve problems. Include
  equations arising from linear and quadratic
  functions, and simple rational and exponential
  functions.
• Write a function defined by an expression in
  different but equivalent forms to reveal and
  explain different properties of the function.
• A-REI.11 Explain why the x-coordinates of the
  points where the graphs of the equations y f(x)
  and y g(x) intersect are the solutions of the
  equation f(x) g(x) find the solutions
  approximately, e.g., using technology to graph
  the functions, make tables of values, or find
  successive approximations. Include cases where
  f(x) and/or g(x) are linear, polynomial,
  rational, absolute value, exponential, and
  logarithmic functions.
• F-IF.7a Graph linear and quadratic functions and
  show intercepts, maxima, and minima.
• F-IF.8a Use the process of factoring and
  completing the square in a quadratic function to
  show zeros, extreme values, and symmetry of the
  graph, and interpret these in terms of a context.
• A-REI.4b Solve quadratic equations by inspection
  (e.g., for x2 49), taking square roots,
  completing the square, the quadratic formula and
  factoring, as appropriate to the initial form of
  the equation. Recognize when the quadratic
  formula gives complex solutions and write them as
  a bi for real numbers a and b.

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Bell Ringer

• Is a value of the input X that makes the output
  f(x) equal zero.
• The zeros of a function are the x-intercepts

What are the zeros of a function ?
Memorize!

• Zero-Product Property
• If ab 0, then a 0 or b 0 Example If (x
  3)(x-7) 0, then (x 3) 0 or (x - 7) 0.

Suggestion Make poster
First sign Second sign ( ) ( )

- - -
- -
- - -
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Solve by Factoring
Product of ac
x 6x -8
2
set to 0 ?
a 1 ?
x 6x 8 0
2
Add (S)
Multiply (P)
S
P
1 8
9
6
Sum of ab
2 4
(x 2)(x 4)
Sign!
F O I L
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FACTORING
x 3x -2
set to 0 ?
2
x 3x 2 0
2
a 1 ?
P S
(x 1)(x 2)
1 2
3
zeros
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Factor example(s) your turn-use dry erase or
note paper

• x2 8x 7 x2 6x 4 x2 12x 32

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Example... Changing the signs

• Factor x2 17x 72
• Reminder find factors with product ac and a b.

( - ) ( - )
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Factoring examples, your turn

• x2 7x 12 x2 11x 24 x2 14x - 32

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• x2 3x 10

( ) ( - )
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Putting it all together

• x2 11x - 15
• Write in standard form
• Factor
• Use the zero-product property
• Solve for x

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Teaching note

• Make factoring worksheet
• Factor
• Find zeros
• Show work

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Exit Question(s)

• Factor x2 5x - 6
• Factor x2 7x 10
• Factor x2 3x 2
• Pick up Factoring worksheet BEGIN! We work
  bell-to-bell
• Worksheet due next class

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Factoring with Technology

• x2 5x - 6
• x2 5x 6
• x2 5x 6

Bell Ringer
Factor and apply Zero-Product Property
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In the next example, you must set the equation
equal to zero before applying technology.
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To solve, simply set the individual factors
equal to zero.
The solutions are -3 and 1/2.
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In this example, you must first factor the
equation. Notice the familiar pattern.

• Factoring Perfect Squares

Factor using difference of two squares.
Notice sign
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Factoring Perfect Squares
2
2
2

• 81x - 36 16x - 6561 4x 25

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Exit Question

• Compare these two functions use a Venn Diagram

2
X 9 0 X 9 0
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Teaching note

• Factoring summation next
• Possibly use as quiz