Unit 1

Functions

Warm Up

In Math 20-1, you learned about reciprocal functions. Reciprocal functions are a special case of rational functions. Reviewing reciprocal functions will help you with this Lesson.
  1. Open Reciprocal of a Function. Adjust the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»m«/mi»«/mstyle»«/math»- and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»b«/mi»«/mstyle»«/math»-values, and watch what happens.
a.
How are the equations of the two graphs related?
b.
Is this relationship also true for the quadratic function and its reciprocal?
  1. Turn on “show vertical asymptotes”.

a.
How are the vertical asymptotes related to each of the functions?
b.
Is this relationship true for both linear and quadratic functions and their reciprocals?

  1. Turn off “show vertical asymptotes” and turn on “show «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«mo»=«/mo»«mo»§#177;«/mo»«mn»1«/mn»«/mstyle»«/math»”.

a.
What occurs at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«mo»=«/mo»«mo»§#177;«/mo»«mn»1«/mn»«/mstyle»«/math»?
b.
Why is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«mo»=«/mo»«mo»§#177;«/mo»«mn»1«/mn»«/mstyle»«/math» significant for reciprocal functions?


a.
For any «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-value, the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-values of the two functions are reciprocals.
b.
Yes, this is true for any function and its reciprocal.

a.
The vertical asymptote of the reciprocal function crosses the graph of the original function at its «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-intercept. For the reciprocal function, the vertical asymptote is a line the graph approaches, but never touches.
b.
Yes, this is true for all functions and their reciprocals.


a.
The graphs of a function and its reciprocal always intersect at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«mo»=«/mo»«mo»§#177;«/mo»«mn»1«/mn»«/mstyle»«/math».
b.
This is where the function and its reciprocal are equal. The reciprocal of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math» is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math» and the reciprocal of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»-«/mo»«mn»1«/mn»«/mstyle»«/math» is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»-«/mo»«mn»1«/mn»«/mstyle»«/math».

If you were unable to confidently complete the activities in this activity,

  • watch the videos below,
  • enter “reciprocal functions” into a search engine to research reciprocal functions, or
  • contact your teacher.

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