Download Notes: Quadratic Functions Vertex Form

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Notes: Quadratic Functions
Graphing from Vertex form
Vertex Form
Vertex:
Axis of Symmetry:
The shape of a quadratic function is a _____________________________.
When graphing you can look at the equation and determine key points.
To find:
1. How do you determine if the graph opens up or down?
2. To find the vertex:
3. To find the axis of symmetry:
4. To find the y-intercept:
5. To find the x-intercepts:
6. Domain:
Range:
7. State the transformations from the parent function.
8. To write in standard form:
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Examples: Identify the vertex.
1. f(x) = 2(x + 1)2 + 4
2. f(x) = (x - 3)2 – 5
3. f(x) = -3(x + 4)2 - 6
4. Given: f(x) = (x - 2)2 - 4
Complete the following:
a. Open up or down: ______
b. Vertex: _______
c. AOS: _______
d. Y-intercept (x = 0): _______
e. Roots (y = 0): __________
f. Transformation: _________________________
g. Standard form: __________________________
h. Sketch graph
Writing Equations given the vertex.
Steps:
1. Plug in __________________________ and the given _____________________________.
2. Solve for _____________.
3. Go back to _______________ _______________, plug in the ______________ and ______ value.
4. Check.
Examples: Write a quadratic equation that contains the given vertex and point.
1. Vertex (-3, 5), Point (-2, 7)
MATH III_ NOTES QUADRATIC FUNCTIONS
2. Vertex (-2, -7), Point (4, -16)
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Graphing from Standard form
Standard form:
Axis of Symmetry:
Vertex:
y-intercept:
Examples:
A.
B.
Direction of opening?: ________________
Direction of opening?: ________________
AOS: ____________
AOS: ____________
Vertex: _______________
Vertex: _______________
y-intercept (x=0): _____________
y-intercept (x=0): _____________
Transformations: ___________________________
Transfor
mations:
___________________
____
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Graphing from Factored form
Factored (Intercept) form:
Roots:
y-intercept:
Examples:
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Writing Equations from Graphs
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Examples:
Imaginary & Complex Numbers
Definition:
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Simplify
1.
9
2.
3  16
3.
6 9
4.
 24
5.
 18
6.
5  72
Definition of complex numbers
where a is a real number and
is imaginary
Simplify
7.
3  4
6
8.
8  12
6
9.
12  9
6
10.
5  100
20
11. (8 – i) + (5 + 4i)
12. (7 – 6i) + (3 – 4i)
13. 10 – (6 + 7i) + 8i
14. (3 – 2i) – (8 – 5i)
Solving Quadratics
Vertex Form (Square Root Method)
Steps:
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1.
2.
3.
4.
5.
Examples.
1. Find the roots : f(x) = (x + 3)2 – 9
2.
Solve: f(x) = 1/2(x - 4)2 – 10
3. Find the roots: f(x) = -2(x + 5)2 + 6
Method #1: Graphing (Look where it crosses the x-axis)
Examples.
1.
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Method #2: Factoring
Steps.
1.
2.
3.
4.
5.
Examples.
1. f(x) = x2 – 16
2. 3. f(x) = 2x2 + 11x + 5
3. 4. 2x2 + 7x = 15
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Method #3: Quadratic Formula
x 
b  b 2  4ac
2a
Examples.
1.
2.
3.
4.
The Discriminant is _______________________. It can tell you about the number of roots and the
nature of those roots.
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Method #4: Completing the Square (CTS)
Steps:
(1)
Move the “c” to the other side of the equation.
(2)
Insert ( ) around the first & second terms.
(3)
Factor the “a” value from the x2 & x terms only.
(4)
Take ½ of the middle term and square it.
(5)
Add this number to both sides of the equation.
(6)
Factor the perfect square trinomial on the left side. CLT on the right side.
Examples:
1.
Solve by completing the square: x2 + 10x + 6 = 0
2.
Write the quadratic equation in vertex form: y = -3x2 + 6x - 11
Writing Quadratic Equations Given the Roots.
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Use standard form
Steps:
1.
2.
3.
Examples:
Write the equation of a quadratic given the roots.
1. x= 1, 8
2. x= 1/3, 2
More on Complex Numbers
To multiply binomial complex numbers, use the definition
and FOIL:
Examples:
1.
2.
To divide by a complex number, first multiply the dividend and divisor by the
complex conjugate of the divisor.
are complex conjugates. The
product of complex conjugates is a real number.
Example:
Examples:
1.
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3.
5.
MATH III_ NOTES QUADRATIC FUNCTIONS
4.
6.
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