L1.3 A1 Rational Functions - Part 6
Completion requirements
Unit 1
Functions
The beginning of this lesson presented a linear function and its related rational function. The related rational function is called a reciprocal function.
Consider the function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mi»x«/mi»«mo»+«/mo»«mn»6«/mn»«/mrow»«/mstyle»«/math» and the reciprocal function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«mo»=«/mo»«mfrac»«mn»1«/mn»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mfrac»«/mstyle»«/math». Discuss the characteristics of the graphs of the two functions, including similarities and differences.
The graph of the function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mi»x«/mi»«mo»+«/mo»«mn»6«/mn»«/mrow»«/mstyle»«/math» will have an «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-intercept at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mo»-«/mo»«mn»6«/mn»«/mrow»«/mstyle»«/math».
The graph of the reciprocal function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«mo»=«/mo»«mfrac»«mn»1«/mn»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»6«/mn»«/mrow»«/mfrac»«/mstyle»«/math» has a vertical asymptote at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mo»-«/mo»«mn»6«/mn»«/mrow»«/mstyle»«/math».
As «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» approaches «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»-«/mo»«mn»6«/mn»«/mstyle»«/math» from the left, the linear function increases and the reciprocal function decreases.
As «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» approaches «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»-«/mo»«mn»6«/mn»«/mstyle»«/math» from the right, the linear function decreases and the reciprocal function increases.
The graph of the reciprocal function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«mo»=«/mo»«mfrac»«mn»1«/mn»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»6«/mn»«/mrow»«/mfrac»«/mstyle»«/math» has a vertical asymptote at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mo»-«/mo»«mn»6«/mn»«/mrow»«/mstyle»«/math».
As «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» approaches «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»-«/mo»«mn»6«/mn»«/mstyle»«/math» from the left, the linear function increases and the reciprocal function decreases.
As «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» approaches «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»-«/mo»«mn»6«/mn»«/mstyle»«/math» from the right, the linear function decreases and the reciprocal function increases.
Points on the graph of a linear function are of the form «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mi»x«/mi»«mo»,«/mo»«mo»§#160;«/mo»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mfenced»«/mstyle»«/math», while points on the graph of the related reciprocal function are of the form «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mi»x«/mi»«mo»,«/mo»«mo»§#160;«/mo»«mfrac»«mn»1«/mn»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mfrac»«/mrow»«/mfenced»«/mstyle»«/math». As such, wherever the graph of a linear function is increasing, the graph of the reciprocal function will be decreasing. Similarly, wherever the graph of the linear function is decreasing, the graph of the reciprocal function will be increasing.
And, wherever there is an «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-intercept on the graph of a linear function, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mi»x«/mi»«mo»,«/mo»«mo»§#160;«/mo»«mn»0«/mn»«/mrow»«/mfenced»«/mstyle»«/math», the related reciprocal function will have a vertical asymptote at the same location because the corresponding point would be «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mi»x«/mi»«mo»,«/mo»«mo»§#160;«/mo»«mfrac»«mn»1«/mn»«mn»0«/mn»«/mfrac»«/mrow»«/mfenced»«/mstyle»«/math».
And, wherever there is an «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-intercept on the graph of a linear function, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mi»x«/mi»«mo»,«/mo»«mo»§#160;«/mo»«mn»0«/mn»«/mrow»«/mfenced»«/mstyle»«/math», the related reciprocal function will have a vertical asymptote at the same location because the corresponding point would be «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mi»x«/mi»«mo»,«/mo»«mo»§#160;«/mo»«mfrac»«mn»1«/mn»«mn»0«/mn»«/mfrac»«/mrow»«/mfenced»«/mstyle»«/math».