L1 Symmetry and Asymptotes - Practice 1
Completion requirements
Unit 3
Curve Sketching
Lesson 1: Symmetry and Asymptotes
Practice
Once you feel confident with symmetry, click on the Practice tab and complete problems 1 and 2. Check your answers by going to the Solutions tab.
Instructions: Click the Download File button to download a printable PDF of the questions. Answer each of the following practice questions on a separate piece of paper. Step by step solutions are provided under the Solutions tab. You will learn the material more thoroughly if you complete the questions before checking the answers.
Instructions: Click the Download File button to download a printable PDF of the questions. Answer each of the following practice questions on a separate piece of paper. Step by step solutions are provided under the Solutions tab. You will learn the material more thoroughly if you complete the questions before checking the answers.
1.
Determine the type of symmetry, if any, in the following graphs.
a.
b.
c.
2.
Determine the type of symmetry, if any, in the graphs of the following functions.
a.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»§#8722;«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mstyle»«/math»
b.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«msup»«mi»x«/mi»«mn»5«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«mi»x«/mi»«/mrow»«/mstyle»«/math»
c.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mfrac»«mrow»«mn»1«/mn»«mo»+«/mo»«mi»x«/mi»«mo»§#8722;«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«mrow»«mi»x«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»
1.
Determine the type of symmetry, if any, in the following graphs.
a.
b.
c.
a.
Since this graph is neither symmetric about the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-axis nor the origin, the function represented has no symmetry.
b.
Since this graph is symmetric about the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-axis, it represents an even function.
c.
Since this graph is neither symmetric about the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-axis nor the origin, the function represented
has no symmetry.
2.
Determine the type of symmetry, if any, in the graphs of the following functions.
a.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»§#8722;«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mstyle»«/math»
b.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«msup»«mi»x«/mi»«mn»5«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«mi»x«/mi»«/mrow»«/mstyle»«/math»
c.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mfrac»«mrow»«mn»1«/mn»«mo»+«/mo»«mi»x«/mi»«mo»§#8722;«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«mrow»«mi»x«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»
a.
Substitute «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»-«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math» for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» in «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»§#8722;«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mstyle»«/math».
If «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mi»f«/mi»«mi
mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math», the function is even. If «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi
mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mo»§#8722;«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math»,
the function is odd. If neither are true, the function has no symmetry.
Since «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math», «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»§#8722;«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mstyle»«/math» is an even function and its graph will be symmetric about the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-axis.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»§#8722;«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mtd»«/mtr»«mtr»«mtd»«mi»f«/mi»«mi
mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mfenced»«mrow»«mo»§#8722;«/mo»«mi»x«/mi»«/mrow»«/mfenced»«mn»4«/mn»«/msup»«mo»§#8722;«/mo»«msup»«mfenced»«mrow»«mo»§#8722;«/mo»«mi»x«/mi»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»§#8722;«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»f«/mi»«mi
mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Since «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math», «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»§#8722;«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mstyle»«/math» is an even function and its graph will be symmetric about the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-axis.
b.
Substitute «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»-«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math» for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» in «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«msup»«mi»x«/mi»«mn»5«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«mi»x«/mi»«/mrow»«/mstyle»«/math».
If «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mi»f«/mi»«mi
mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math», the function is even. If «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi
mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mo»§#8722;«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math»,
the function is odd. If neither are true, the function has no symmetry.
Since «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mo»§#8722;«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math», «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«msup»«mi»x«/mi»«mn»5«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«mi»x«/mi»«/mrow»«/mstyle»«/math» is an odd function and its graph will be symmetric about the origin.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mi»x«/mi»«mn»5«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«mi»x«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»f«/mi»«mi
mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mfenced»«mrow»«mo»§#8722;«/mo»«mi»x«/mi»«/mrow»«/mfenced»«mn»5«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«mfenced»«mrow»«mo»§#8722;«/mo»«mi»x«/mi»«/mrow»«/mfenced»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«msup»«mi»x«/mi»«mn»5«/mn»«/msup»«mo»§#8722;«/mo»«mn»3«/mn»«mi»x«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mi
mathvariant=¨normal¨»(«/mi»«msup»«mi»x«/mi»«mn»5«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi
mathvariant=¨normal¨»)«/mi»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Since «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mo»§#8722;«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math», «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«msup»«mi»x«/mi»«mn»5«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«mi»x«/mi»«/mrow»«/mstyle»«/math» is an odd function and its graph will be symmetric about the origin.
c.
Substitute «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»-«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math» for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» in «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mfrac»«mrow»«mn»1«/mn»«mo»+«/mo»«mi»x«/mi»«mo»§#8722;«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«mrow»«mi»x«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math».
If «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mi»f«/mi»«mi
mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math», the function is even. If «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi
mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mo»§#8722;«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math»,
the function is odd. If neither are true, the function has no symmetry.
Since «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»§#8800;«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»§#8800;«/mo»«mo»§#8722;«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math», the function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mfrac»«mrow»«mn»1«/mn»«mo»+«/mo»«mi»x«/mi»«mo»§#8722;«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«mrow»«mi»x«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math» is neither even nor odd. As such, the graph of the function has no symmetry.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»1«/mn»«mo»+«/mo»«mi»x«/mi»«mo»§#8722;«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«mrow»«mi»x«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mi»f«/mi»«mi
mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»1«/mn»«mo»+«/mo»«mfenced»«mrow»«mo»§#8722;«/mo»«mi»x«/mi»«/mrow»«/mfenced»«mo»§#8722;«/mo»«msup»«mfenced»«mrow»«mo»§#8722;«/mo»«mi»x«/mi»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mrow»«mrow»«mfenced»«mrow»«mo»§#8722;«/mo»«mi»x«/mi»«/mrow»«/mfenced»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»1«/mn»«mo»§#8722;«/mo»«mi»x«/mi»«mo»§#8722;«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«mrow»«mo»§#8722;«/mo»«mi»x«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»§#8800;«/mo»«/mtd»«mtd»«mi»f«/mi»«mi
mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»§#8800;«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Since «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»§#8800;«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»§#8800;«/mo»«mo»§#8722;«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math», the function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mfrac»«mrow»«mn»1«/mn»«mo»+«/mo»«mi»x«/mi»«mo»§#8722;«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«mrow»«mi»x«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math» is neither even nor odd. As such, the graph of the function has no symmetry.