Name ____________________________________ Date _________________________ Block ________
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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Name ____________________________________ Date _________________________ Block ________
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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Name ____________________________________ Date _________________________ Block ________
Graphing from Standard form
Standard form:
Axis of Symmetry:
Vertex:
y-intercept:
Examples:
A. 𝑓(𝑥) = 𝑥 2 + 2𝑥 − 6
B. 𝑓(𝑥) = 2𝑥 2 + 8𝑥 − 3
Direction of opening?: ________________
Direction of opening?: ________________
AOS: ____________
AOS: ____________
Vertex: _______________
Vertex: _______________
y-intercept (x=0): _____________
y-intercept (x=0): _____________
Transformations: ___________________________
Transformations: _______________________
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Name ____________________________________ Date _________________________ Block ________
Graphing from Factored form
Factored (Intercept) form:
Roots:
y-intercept:
Examples:
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Name ____________________________________ Date _________________________ Block ________
Writing Equations from Graphs
Examples:
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Imaginary & Complex Numbers
Definition:
𝑖 = √−1
𝑖 3 = −𝑖
𝑖 2 = −1
𝑖4 = 1
Simplify
1.
9
2.
3  16
3.
6 9
4.
 24
5.
 18
6.
5  72
Definition of complex numbers: 𝑎 + 𝑏𝑖 where a is a real number and 𝑏𝑖 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)
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Name ____________________________________ Date _________________________ Block ________
Solving Quadratics
Vertex Form (Square Root Method)
Steps:
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
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Name ____________________________________ Date _________________________ Block ________
Method #1: Graphing (Look where it crosses the x-axis)
Examples.
1.
2.
Method #2: Factoring
Steps.
1.
2.
3.
4.
5.
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Name ____________________________________ Date _________________________ Block ________
Examples.
1. f(x) = x2 – 16
2. 3. f(x) = 2x2 + 11x + 5
3. 4. 2x2 + 7x = 15
Method #3: Quadratic Formula
b  b 2  4ac
x 
2a
Examples.
1.
2.
3.
4.
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Name ____________________________________ Date _________________________ Block ________
The Discriminant is _______________________. It can tell you about the number of roots and the
nature of those roots.
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
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Name ____________________________________ Date _________________________ Block ________
Writing Quadratic Equations Given the Roots.
Use standard form 𝒚 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄
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 𝒊𝟐 = −𝟏 and FOIL:
Examples:
1.
2.
To divide by a complex number, first multiply the dividend and divisor by the
complex conjugate of the divisor. 𝒂 + 𝒃𝒊 𝒂𝒏𝒅 𝒂 − 𝒃𝒊 are complex conjugates. The
product of complex conjugates is a real number.
Example:
Examples:
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Name ____________________________________ Date _________________________ Block ________
1.
3.
5.
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2.
4.
6.
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