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5.1 Use Transformations to
Graph Quadratic Functions
Example 1:
y
Example 2:
x
y
x
1
Example 3:
y
x
2
Example 4:
a.
b.
Transformation: ____________________ Transformation: ____________________
y
y
x
x
c. g(x) = ( x – 4)² + 3
d.
Transformation: ____________________ Transformation: ____________________
y
y
x
x
f. g(x) = (x ─ 5)² + 4
e. g(x) = ( x + 2)² + 1
Transformation: ____________________ Transformation: ____________________
y
y
x
x
3
Vertex of a parabola: ________________________________________________________
Vertex form: _______________________________________________________________
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Example 5:
a.
b.
y
y
x
c.
x
d.
y
y
x
x
e.
f.
y
y
x
x
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Vertex Form: __________________________________
Example 6: Use the description to write the quadratic function g(x) in vertex form.
a. The parent function f(x) = x² is shifted right 3 units, and up 4 units.
c. The parent function f(x) = x² is shifted left 2 units, and down 5 units.
d. The parent function f(x) = x² is shifted right 1 units, and up 3 units.
e. The parent function f(x) = x² is reflected over the x The axis, shifted left 6 units, and down
2 units.
f. The parent function f(x) = x² is reflected over the x axis, shifted right 1 unit, and down 7
units.
Example 7: Describe the transformation from the parent function f(x) = x²
a. g(x) = (x – 3)² + 1 _________________
b. g(x) = – (x + 2)² – 4 ______________
c. g(x) = (x + 1)² + 2 __________________
c. g(x) = (x – 5)² – 3 ______________
e. g(x) = – (x + 3)² – 5 _________________
f. g(x) = (x + 1)² – 7 ______________
g. g(x) = (x + 3)² + 8 __________________
h. g(x) = – (x – 5)² + 6 ______________
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The axis of symmetry is always the vertical line x = a number
(the number is the ___________________ of the vertex)
Example 1:
a. Identify the axis of symmetry and vertex for the graph f(x) = 2(x + 2)² – 3,
then draw the graph.
y
x
Identify the axis of symmetry and vertex for the graphs, then draw the graphs.
b.
c.
y
y
x
x
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Standard form of a quadratic equation: ___________________________________________________
Where a, b, c are real numbers and a ≠ 0.
If the equation is in standard form, the parabola has the following properties:
The equation opens upward if a is ______________________.
The equation opens downward if a is ____________________.
The equation for the axis of symmetry is ________________.
The y intercept is the ________ value.
The vertex is the point (
,
)
Example 3:
a.
b.
y
c.
d.
x
e.
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Example 4:
a.
b.
c.
y
d.
x
e.
Example 5:
a.
b.
y
y
x
x
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When a parabola opens upward, the y-value of the vertex is the __________________ value.
When a parabola opens downward, the y-value of the vertex is the __________________ value.
Example 6:
a.
c.
b.
d.
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5.3
Zeros of a function: _________________________________________________________________
Example 1:
a.
b.
y
y
x
x
c.
y
x
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Factor.
1. x² + 7x + 6
2. x² – x – 2
3. x² – 5x + 4
4. x² – 7x + 10
5. x² + x –12
6. x² –2x + 1
7. x² + 3x – 10
8. x² + 5x + 4
9. x² + 6x + 5
.
15. 5x² – 15x
16. 2x² – 16x + 30
17. – x² + x + 56
18. 3x² + 18x +27
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When a polynomial has __________ terms, you can make two groups and factor
out the __________ of each group.
Factor by grouping.
1. 12x³ – 9x² + 20x – 15
3. 2x³ – 2x² – 3x + 3
2. 6x³ – 8x² – 15x + 20
4. 6x³ + 18x² + x + 3
To
Factoring ax² + bx + c, when a ≠ 1
“AC Method”
Step 1: Multiply a∙c
Together:
Step 2: What factors of a∙c combine to b?
6x² + 11x + 3
Step 3: Write as a 4 term polynomial
Step 4: Factor by grouping
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Factor completely.
a. 4x² + 16x + 15
d. 9x² – 15x + 4
g. 5x² + 11x + 2
j. 2x² +13x + 15
b. 2x² + 11x + 12
e. 3x² + 13x + 12
h. 10x² – 9x – 1
k. 3x² + 10x + 8
c. 5x² – 14x + 8
f. 7x² – 3x – 10
i. 15X² + 4X – 3
l. 2x² – 7x –15
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The solutions to a quadratic equation of the form ax² + bx + c = 0 are called _____________.
The roots of an equation are the values of the variable that make the equation ___________.
You can find the roots of some quadratic equations by ________________ and applying the
______ product property.
Example 2:
a.
b. f(x) = x² + 6x – 7
c. f(x) = x² – 3x – 28
d. g(x) = 4x² + 8x
e. g(x) = 10x² – 25x
f. g(x) = 6x² – 9x
g.
h.
j. f(x) = 3x² + 8x + 4
k. f(x) = 2x² – 11x + 5
i.
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Make a list of the first 15 square numbers:
Difference of two squares:
a² ─ b² = (
)(
)
Example 4:
a.
b. 36x² = 49
c. 36x² = 100
d.
e. 4x² = 25
f. 81x² = 144
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Example 5:
a.
b.
c.
d. Write a quadratic function in standard form with zeros 6 and 2.
e. Then Write a quadratic function in standard form with zeros -3 and -2.
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