Download Ch. 4 Notes

Chapter 4 Quadratic Functions and Factoring
Foil:
Factoring:
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Opposite of Foil ( multiply: first, outside, inside, last terms)
Factor out into two expressions
Determine what numbers multiply together for last term (c), but add together for middle
term (b)
● If the last term is (+), the signs are the same, the x term tells you which sign it is
● If the last term is (­), the signs are opposite, put sign (+/­) of x term with the higher
number
*When there is a number in front of the coefficient, you will need to factor out. Can use box, and
super box method.
* May need to pull out a Greatest common factor that goes into the entire equation to make
factoring easier.
* When solving quadratics by factoring, need to set the equation to zero, before you can begin
factoring. (Get it in standard form ax 2 + bx + c = 0 )
Steps: (using the box/super box method of factoring)
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Set up an X shape
Put A * C Value on top
Put B value on bottom
Determine what numbers add to be B value, multiply to be A*C value
Create a box with Ax^2 value in top left, C value in bottom right, and the factors in either
box with X.
● Factor out what each box has in common, going across twice, and up twice.
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Create two factor groups, and set to zero. Solve to find roots.
Zero Product Property
If ab = 0
Then a = 0 and/or b = 0
Properties for Radicals:
√ab = √a ∙ √b
a
√a √ b = √b (√x ) 2 = x √x 2 = x
* Perfect squares ( number times itself): 4,9,16,25,36,49,64,81,100,121,144,169,196,225
Rules for Reducing Radicals:
1. There can be no perfect square factors under the radical sign (square root).
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(insert example) Determine what biggest perfect square is a factor.
Break into two radicals: one a perfect square, one it's multiplicate.
Pull out the number from the perfect square root.
Put back together.
2. There can be no fractions under the radical.
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Keep reducing until all of the rules are met.
3. There can be no radicals in the denominator.
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Multiply top and bottom of fraction by the bottom value, to eliminate the radical in
the denominator
Solving with square root method:
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Whenever you take the square root of both sides of an equation to solve, remember
plus/and minus symbol.
● BIG IDEA: to get rid of x squared, take the square root of it, as well as the other side of
the equation.
● Before solving, divide by the greatest common factor.
● May need to take a few steps, to isolate the squared term.
Solving by Completing the Square:
Steps:
1. Subtract C from both sides and leave a space for a new constant
2. Pick a value to add to both sides to make it a perfect square do this by dividing B by 2,
and squaring the result. Add this to both sides (in the space).
3. Rewrite the left side as ( x + B/2 ) squared. (the third term is no longer used on the left)
4. Solve using the square root method ­ square root both sides to make x squared
disappear.
5. Reduce to lowest form
Solve by using the Quadratic Formula:
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Can be used when the equation is not able to be factored
Must be written in standard form = 0
Determine what numbers are used for A, B, C, plug into the equation
Formula easiest to use if there is a value in front of A rather than completing the square
Solve the discriminant first (the info under the radicand) tells how many real or imaginary
solutions exist.
Rules:
1. If B^2 ­ 4ac > 0 , two real solutions
2. If B^2 ­ 4ac = 0 , one real solution
3. If B^2 ­ 4ac < 0 , no real solutions, 2 imaginary solutions (all solutions)
Vertical Motion Model:
Formula: h = ­16t^2 + vt+ s
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h = final height in feet
v = initial velocity
­16 = gravity
s = starting height
t = time in seconds
Example: Drop a rock from 200 feet in the air, how long does it take for it to hit the ground?
0 = ­16t^2 + 200
Imaginary Unit:
BIG IDEAS:
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When adding or subtracting factors including i, line up like terms and solve
When multiplying factors, use FOIL, and remember to change i^2 to ­1 and then combine
like terms.
● Whenever there is a ­1 under radical sign, remember to change to i.
● When reducing to lowest terms:
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Use FOIL to simplify
Convert i^2 to ­1
Combine like terms
Graphing quadratic functions in standard form standard form ax 2 + bx + c = 0
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If (A) is positive, the parabola opens up, if (A) is negative, it opens down
The X coordinate of the vertex is ­b/2a
Axis of symmetry is the vertical line = ­b/2a
● To Sketch:
1. Find the X coordinate of
the vertex (x,y). Plug X in to get
the Y coordinate. Plot the Vertex.
2. Construct table of values,
using two X values to the left and
right of the vertex
3. Plot the points and
connect them with a parabola. (
They should reflect over the y
axis)
* Wherever the parabola
touches/crosses the X intercept,
there is a real solution.
● touches in 2 points = 2
solutions
● If vertex is only point that
touches X = 1 solution
● If parabola does not touch
the X = no solutions
* This graph does not go with
the above example. It is just a
picture.
Graphing Quadratics in Vertex Form
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K is the y value, Opposite of h is the x value.
Inside the ( ) effects the x values ­ moves the opposite of what you think ­ from the vertex
Outside the () effects the y values ­ moves the way you think ­ from the vertex
Finding Minimum and Maximum Values ( Only the y axis value)
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The maximum or minimum is the value of the function at .
Whether it is a maximum or minimum depends on whether the parabola opens up =
minimum, or down = maximum.
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You can tell which way the parabola opens by the sign on the lead coefficient. if is down, if , it isup, and, of course, if you don't have a parabola at all.
● If the vertex is the top of the graph, the vertex has the maximum y value.
● If the vertex is the bottom of the graph, it has the minimum y value.
, it