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Chapter 2Section 2.7
Graphing Techniques
Multiplying the Function or the Independent Variable by a Number
6 5 4 3 2 1 1 2 3 4 5 6
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6f x
x-intercepts of this function are at the points:
6 5 4 3 2 1 1 2 3 4 5 6
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1
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6 f x
Multiplying by reflects graph about x-axis.
x-intercepts of this function are at the points:
6 5 4 3 2 1 1 2 3 4 5 6
6
5
4
3
2
1
1
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6f x
Multiplying by reflects graph about y-axis.
x-intercepts of this function are at the points:
Symmetry about the x and y AxesA graph is symmetric about the x-axis if you reflect is graph about the x-axis it dos not change this means that ():
Replacing x by –x in the equation does not change the equation!
A graph is symmetric about the y-axis if you reflect is graph about the y-axis it dos not change this means that:
Replacing y by –y in the equation does not change the equation!
replace x by –x : This is symmetric about the x-axis
replace x by –x : This is not symmetric about the x-axis
replace y by –y : This is symmetric about the y-axis
replace y by –y : This is not symmetric about the y-axis
Multiplying the Function or the Independent Variable by a Number
6 5 4 3 2 1 1 2 3 4 5 6
6
5
4
3
2
1
1
2
3
4
5
6f x
x-intercepts of this function are at the points:
Multiplying by a number greater than 1 stretches it vertical.
x-intercepts of this function are at the points:
Multiplying by a number less than 1 shrinks it vertical.
x-intercepts of this function are at the points:
7 6 5 4 3 2 1 1 2 3 4 5 6 7
7
6
5
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73 f x
6 5 4 3 2 1 1 2 3 4 5 6
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12f x
Multiplying the Function or the Independent Variable by a Number
x-intercepts of this function are at the points:
Multiplying by a number greater than 1 shrinks it horiz.
x-intercepts of this function are at the points:
Multiplying by a number less than 1 stretches it horiz.
x-intercepts of this function are at the points:
7 6 5 4 3 2 1 1 2 3 4 5 6 7
7
6
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1
1
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7f x
4 3 2 1 1 2 3 4
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4f 2 x
7 6 5 4 3 2 1 1 2 3 4 5 6 7
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f 12x
Multiplying the Function or the Independent Variable by a Number
x-intercepts of this function are at the points:
Multiplying and by a number does both.
x-intercepts of this function are at the points:
Multiplying by a number less than 1 stretches it horiz.
x-intercepts of this function are at the points:
7 6 5 4 3 2 1 1 2 3 4 5 6 7
7
6
5
4
3
2
1
1
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5
6
7f x
7 6 5 4 3 2 1 1 2 3 4 5 6 7
7
6
5
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2
1
1
2
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5
6
7
12f x
7 6 5 4 3 2 1 1 2 3 4 5 6 7
7
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1
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3 f12x
Translating a Function Left or Right
x-intercepts of this function are at the points:
7 6 5 4 3 2 1 1 2 3 4 5 6 7
7
6
5
4
3
2
1
1
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7f x
7 6 5 4 3 2 1 1 2 3 4 5 6 7
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7f x 4
Adding to shifts the graph to the left by that number.
x-intercepts of this function are at the points:
7 6 5 4 3 2 1 1 2 3 4 5 6 7
7
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5
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2
1
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7f x 2
Subtracting from shifts the graph to the right by that number.
x-intercepts of this function are at the points:
Translating a Function Up or Down
x-intercepts of this function are at the points:
7 6 5 4 3 2 1 1 2 3 4 5 6 7
7
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5
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1
1
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7f x
Adding to shifts the graph to the up by that number.
The graph now intersects the line at the points:
Subtracting from shifts the graph to the down by that number.
The graph now intersects the line at the points:
7 6 5 4 3 2 1 1 2 3 4 5 6 7
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1
1
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7f x 5
7 6 5 4 3 2 1 1 2 3 4 5 6 7
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7f x 2
Combining Graphical OperationsFind the graph of the function by breaking in a series of transformations of .
Start with:Reflect x-axis:Stretch by 2:Move up 3:
7 6 5 4 3 2 1 1 2 3 4 5 6 7
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73 2 x
7 6 5 4 3 2 1 1 2 3 4 5 6 7
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Find the equation for the graph above:
Start with:Move right 2:Move down 5:
