171S2.4q Symmetry
February 14, 2013
MAT 171 Precalculus Algebra
2.4 Symmetry
Dr. Claude Moore
Cape Fear Community College
CHAPTER 2: More on Functions
2.1 Increasing, Decreasing, and Piecewise Functions; Applications
2.2 The Algebra of Functions
2.3 The Composition of Functions
2.4 Symmetry
2.5 Transformations 2.6 Variation and Applications
Symmetry with respect to the Origin or the Y­axis.
http://cfcc.edu/mathlab/geogebra/function_symmetry.html
• Determine whether a graph is symmetric with respect to the x­axis, the y­axis, and the origin.
• Determine whether a function is even, odd, or neither even nor odd.
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Sep 19­3:27 PM
Sep 19­3:27 PM
Example
Symmetry
Test x = y2 + 2 for symmetry with respect to the x­axis, the y­axis, and the origin.
Algebraic Tests of Symmetry:
x­axis: If replacing y with ­y produces an equivalent equation, then the graph is symmetric with respect to the x­
axis.
x­axis: We replace y with ­y:
x = y2 + 2
x = (­y)2 + 2
x = y2 + 2
The resulting equation is equivalent to the original so the graph is symmetric with respect to the x­axis.
y­axis: If replacing x with ­x produces an equivalent equation, then the graph is symmetric with respect to the y­
axis.
Origin: If replacing x with ­x and y with ­y produces an equivalent equation, then the graph is symmetric with respect to the origin.
y­axis: We replace x with ­x:
x = y2 + 2
(­x) = y2 + 2
­x = y2 + 2
The resulting equation is not equivalent to the original so the graph is not symmetric with respect to the y­axis.
Symmetry with respect to the Origin or the Y­axis.
http://cfcc.edu/mathlab/geogebra/function_symmetry.html
Origin: We replace x with −x and y with −y:
x = y2 + 2
(­x)= (­y)2 + 2
­x = y2 + 2
The resulting equation is not equivalent to the original equation, so the graph is not symmetric with respect to the origin.
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Even and Odd Functions
If the graph of a function f is symmetric with respect to the y­axis, we say that it is an even function. That is, for each x in the domain of f, f(x) = f(­x).
If the graph of a function f is symmetric with respect to the origin, we say that it is an odd function. That is, for each x in the domain of f, f(­x) = ­f(x).
197/1-6.
Determine if symmetric with respect to x­axis y­axis
origin
1.
2.
Example
Determine whether the function is even, odd, or neither.
3.
4.
5.
1. 6.
We see that h(x) = h(­x). Thus, h is even.
Determine whether the function is even, odd, or neither.
2.
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Sep 22­1:22 PM
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171S2.4q Symmetry
197/8. First, graph the equation and determine visually whether it is symmetric with respect to the x­axis, the y­
axis, and the origin. Then verify your assertion algebraically. y = | x + 5 |
Sep 22­1:29 PM
197/12. First, graph the equation and determine visually whether it is symmetric with respect to the x­axis, the y­
axis, and the origin. Then verify your assertion algebraically. x2 + 4 = 3y
Sep 22­1:29 PM
197/18. Test algebraically whether the graph is symmetric with respect to the x­axis, the y­axis, and the origin. Then check your work graphically, if possible, using a graphing calculator. 5y = 7x2 ­ 2x
Sep 22­1:32 PM
February 14, 2013
197/10. First, graph the equation and determine visually whether it is symmetric with respect to the x­axis, the y­
axis, and the origin. Then verify your assertion algebraically. 2x ­ 5 = 3y
Sep 22­1:29 PM
197/14. First, graph the equation and determine visually whether it is symmetric with respect to the x­axis, the y­
axis, and the origin. Then verify your assertion algebraically. y = ­4 / x
Sep 22­1:29 PM
197/22. Test algebraically whether the graph is symmetric with respect to the x­axis, the y­axis, and the origin. Then check your work graphically, if possible, using a graphing calculator. 2y2 = 5x2 + 12
Sep 22­1:32 PM
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171S2.4q Symmetry
February 14, 2013
197/24. Test algebraically whether the graph is symmetric with respect to the x­axis, the y­axis, and the origin. Then check your work graphically, if possible, using a graphing calculator. 3x = | y |
Sep 22­1:32 PM
197/33­38. Determine visually whether the function is even, odd, or neither even nor odd.
197/26. Test algebraically whether the graph is symmetric with respect to the x­axis, the y­axis, and the origin. Then check your work graphically, if possible, using a graphing calculator. xy ­ x2 = 3
Sep 22­1:32 PM
198/40. Test algebraically whether the function is even, odd, or neither even nor odd. Then check your work graphically, where possible, using a graphing calculator. f(x) = 7x3 + 4x ­ 2
even odd neither
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34.
35.
36.
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38.
Sep 22­1:32 PM
198/41. Test algebraically whether the function is even, odd, or neither even nor odd. Then check your work graphically, where possible, using a graphing calculator.
f(x) = 5x2 + 2x4 ­ 1
Sep 22­1:38 PM
Sep 22­1:38 PM
198/42. Test algebraically whether the function is even, odd, or neither even nor odd. Then check your work graphically, where possible, using a graphing calculator.
f(x) = x + 1/x
Sep 22­1:38 PM
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171S2.4q Symmetry
February 14, 2013
198/44. Test algebraically whether the function is even, odd, or neither even nor odd. Then check your work graphically, where possible, using a graphing calculator. f(x) = ∛x Sep 22­1:38 PM
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