L4 Trigonometric Function Graphing Review - Part 7
Completion requirements
Unit 4A
Trigonometry Part 1
Lesson 4: Trigonometric Function Graphing Review
Amplitude
The amplitude of a function’s graph is half the distance from the minimum to the maximum, or the distance between the midline and either the maximum or minimum, as shown in the graphs below.The amplitude of «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»§#952;«/mo»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mi»sin«/mi»«mo»§#952;«/mo»«/mrow»«/mstyle»«/math» is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math».
The amplitude of «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»§#952;«/mo»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mi»cos«/mi»«mo»§#952;«/mo»«/mrow»«/mstyle»«/math» is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math».
Since there are no maximum or minimum values for «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»§#952;«/mo»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mi»tan«/mi»«mo»§#952;«/mo»«/mrow»«/mstyle»«/math», there is no amplitude.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»amplitude«/mi»«mo»=«/mo»«mfrac»«mfenced open=¨|¨ close=¨|¨»«mrow»«mi»max«/mi»«mo»§#8722;«/mo»«mi»min«/mi»«/mrow»«/mfenced»«mn»2«/mn»«/mfrac»«/mrow»«/mstyle»«/math»
The amplitude of a graph corresponds to a vertical stretch. If a trigonometric function is multiplied by a constant «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»a«/mi»«/mstyle»«/math», such as «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»§#952;«/mo»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mi»a«/mi»«mi»sin«/mi»«mo»§#952;«/mo»«/mrow»«/mstyle»«/math», the function is vertically stretched by a factor of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨|¨ close=¨|¨»«mi»a«/mi»«/mfenced»«/mstyle»«/math».
Graph «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»§#952;«/mo»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»2«/mn»«mi»sin«/mi»«mo»§#952;«/mo»«/mrow»«/mstyle»«/math»
and state the amplitude of the function.
Because the sine function is multiplied by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math», all «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-values must be multiplied by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math», causing a vertical stretch of
the sine function by a factor of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math».
The amplitude of «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»§#952;«/mo»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»2«/mn»«mi»sin«/mi»«mo»§#952;«/mo»«/mrow»«/mstyle»«/math» is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math». This can be read from the equation of the function - amplitude is the coefficient of the function. The amplitude can also be calculated using the formula introduced above.
The amplitude of «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»§#952;«/mo»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»2«/mn»«mi»sin«/mi»«mo»§#952;«/mo»«/mrow»«/mstyle»«/math» is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math». This can be read from the equation of the function - amplitude is the coefficient of the function. The amplitude can also be calculated using the formula introduced above.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»amplitude«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mfenced open=¨|¨ close=¨|¨»«mrow»«mi»max«/mi»«mo»§#8722;«/mo»«mi»min«/mi»«/mrow»«/mfenced»«mn»2«/mn»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mfenced
open=¨|¨ close=¨|¨»«mrow»«mn»2«/mn»«mo»§#8722;«/mo»«mfenced»«mrow»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«/mrow»«/mfenced»«mn»2«/mn»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»4«/mn»«mn»2«/mn»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»2«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»