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8.6 1
Section 8.6 Quadratic Functions & Graphs
Graphing Parabolas f(x)=ax2
f(x)=ax2+k f(x)=a(x–h)2
f(x)=a(x–h)2+k Finding the Vertex and Axis of Symmetry Finding Maximums or Minimums
8.6 2
Definition
Let’s make a table of values for f(x) = x2
Then sketch the basic parabola shape …
8.6 3
Make a table of values for f(x) = x2
Then graph it
8.6 4
Similar Curves Changing the
coefficient of x2: Smaller is wider Larger is narrower
The vertex remains the same
8.6 5
Make a table of values for f(x) = -½x2
Then graph it
8.6 6
Reflections The graph of y= -ƒ(x) is the graph of y = ƒ(x)
reflected about the x-axis.
)(xf
8.6 7
Vertical Translations If ƒ is a function and k is a positive number, then The graph of y = ƒ(x) + k is identical to the
graph of y =ƒ(x) except that it is translated k units upward.
The graph of y = ƒ(x) - k is identical to the graph of y = ƒ(x) except that it is translated k units downward.
Sketch f(x) = x2 + 3 on the board
kxf )(
8.6 8
Horizontal TranslationsIf ƒ is a function and h is a positive number, Then the graph of y = ƒ(x - h) is identical to
the graph of y = ƒ(x) except that it is translated h units to the right.
The graph of y = ƒ(x + h) is identical to the graph of y = ƒ(x) except that it is translated h units to the left.
)( hxf
8.6 9
Graph f(x) = (x – 3)2
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Graph g(x) = -2(x +4)2
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khxaxf 2)()(Equation The
8.6 12
Graph g(x) = (x – 3)2 – 5
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Finding the Vertex and Axis of SymmetryAnswer the following questions about the given
equation:
a. Does the graph open upward or downward?
b. What are the coordinates of the vertex?
c. What is the axis of symmetry?
d. Is it narrow or wide?
Up
(5,1)
x=5narrow
156 2 xy
8.6 14
Finding the Vertex and Axis of SymmetryAnswer the following questions about the given
equation:
a. Does the graph open upward or downward?
b. What are the coordinates of the vertex?
c. What is the axis of symmetry?
d. Is it narrow or wide?
Down
(-4,-3)
x=-4wide
34 2
32 xy
8.6 15
Maximums & Minimums
8.6 16
Maximum or Minimum?Vertex and Axis of Symmetry?
4)0,4(0)4()(
3)5,3(55)3(2
2)6,2(66)2()(
7)5,7(55)7()(
232
2
221
2
xAxisyMaxxxf
xAxisyMaxxy
xAxisyMinxxh
xAxisyMinxxg
8.6 17
But what if the function is not in the form of f(x) = a(x – h)2 + k ?
5)3()(
3436)(
46)(
46)(
!
2
222
2
2
xxf
xxxf
xxxf
xxxf
squaretheComplete
1)2(3)(
2313)24(3)(
13)4(3)(
13123)(
!
2
222
2
2
xxf
xxxf
xxxf
xxxf
squaretheComplete
8.6 18
What Next? Section 9.1Composite Functions
Sections 8.7 - 8.9 may not be covered
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