Unit 3

Transformations

Horizontal Stretches (Changes in the Period)

Recall a horizontal stretch results in a change in the shape of the graph of a function. In sinusoidal functions, the horizontal stretch affects the period.

Both the base sine function and the base cosine function have a period of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»2«/mn»«mo mathvariant=¨italic¨»§#960;«/mo»«/mrow»«/mstyle»«/math» or «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»360«/mn»«mo»§#176;«/mo»«/mrow»«/mstyle»«/math». In other words, the cycles of sine and cosine functions repeat every «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»2«/mn»«mo mathvariant=¨italic¨»§#960;«/mo»«/mrow»«/mstyle»«/math» or «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»360«/mn»«mo»§#176;«/mo»«/mrow»«/mstyle»«/math».

A horizontal stretch of a sinusoidal function causes the period to change length. It will either become longer or shorter.

If «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»sin«/mi»«mo»§#952;«/mo»«/mrow»«/mstyle»«/math» undergoes a horizontal stretch, the transformed function is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»sin«/mi»«mfenced open=¨[¨ close=¨]¨»«mrow»«mi»b«/mi»«mfenced»«mo»§#952;«/mo»«/mfenced»«/mrow»«/mfenced»«/mrow»«/mstyle»«/math».

Extending what you learned in Lesson 3.1, a function of the form «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»sin«/mi»«mi»b«/mi»«mfenced»«mo»§#952;«/mo»«/mfenced»«/mrow»«/mstyle»«/math» will undergo a horizontal stretch by a factor of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfrac»«mn»1«/mn»«mfenced open=¨|¨ close=¨|¨»«mi»b«/mi»«/mfenced»«/mfrac»«/mstyle»«/math».

The sine function has a period of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»2«/mn»«mo mathvariant=¨italic¨»§#960;«/mo»«/mrow»«/mstyle»«/math», so a horizontal stretch will affect the period by a factor of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfrac»«mn»1«/mn»«mfenced open=¨|¨ close=¨|¨»«mi»b«/mi»«/mfenced»«/mfrac»«/mstyle»«/math».

Period «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»=«/mo»«mn»2«/mn»«mo mathvariant=¨italic¨»§#960;«/mo»«mfenced»«mfrac»«mn»1«/mn»«mfenced open=¨|¨ close=¨|¨»«mi»b«/mi»«/mfenced»«/mfrac»«/mfenced»«mo»=«/mo»«mfrac»«mrow»«mn»2«/mn»«mo mathvariant=¨italic¨»§#960;«/mo»«/mrow»«mfenced open=¨|¨ close=¨|¨»«mi»b«/mi»«/mfenced»«/mfrac»«/mrow»«/mstyle»«/math»

The same is true for the cosine function.

Note the tangent function has a period of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo mathvariant=¨italic¨»§#960;«/mo»«/mstyle»«/math», so a horizontal stretch will affect the period of the tangent function by a factor of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfrac»«mo mathvariant=¨italic¨»§#960;«/mo»«mfenced open=¨|¨ close=¨|¨»«mi»b«/mi»«/mfenced»«/mfrac»«/mstyle»«/math». In degrees, the tangent function has a period of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»180«/mn»«mo»§#176;«/mo»«/mstyle»«/math», so a horizontal stretch will affect the period of the tangent function by a factor of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfrac»«mrow»«mn»180«/mn»«mo»§#176;«/mo»«/mrow»«mfenced open=¨|¨ close=¨|¨»«mi»b«/mi»«/mfenced»«/mfrac»«/mstyle»«/math».