functions and equations
DESCRIPTION
Functions and equations. Mr. Thauvette DP SL Mathematics. Graphs of Functions. The x -intercepts of a function are the values of x for which y = 0. They are the zeros (i.e., solutions, roots) of the function. The y -intercept of a function is the value of y when x = 0. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/1.jpg)
FUNCTIONS AND EQUATIONS
Mr. ThauvetteDP SL Mathematics
![Page 2: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/2.jpg)
Graphs of Functions
The x-intercepts of a function are the values of x for which y = 0. They are the zeros (i.e., solutions, roots) of the function.
The y-intercept of a function is the value of y when x = 0.
![Page 3: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/3.jpg)
Graphs of Functions
An asymptote is a line that the graph approaches or begins to looklike as it tends to infinity in a particular direction.
vertical asymptote horizontal asymptotey = 2
x = 2
![Page 4: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/4.jpg)
Graphs of Functions
To find vertical asymptotes, look for values of x for which thefunction is undefined:
• If find where
• If find where
To find horizontal asymptotes, consider the behaviour as
![Page 5: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/5.jpg)
Transformations of Graphs• translates vertically units.
• translates horizontally units.
• translates by the vector
![Page 6: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/6.jpg)
• translates vertically units.
![Page 7: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/7.jpg)
• translates vertically units.
![Page 8: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/8.jpg)
• translates vertically units.
![Page 9: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/9.jpg)
• translates vertically units.
![Page 10: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/10.jpg)
Examples
Find the equation of the relation under the translation vectorindicated. Graph both the original and translated relations on the same set of axes.
(a) (b)
![Page 11: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/11.jpg)
Example (a)
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Example (a)
![Page 13: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/13.jpg)
Example (b)
![Page 14: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/14.jpg)
Example (b)
![Page 15: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/15.jpg)
• translates horizontally units.
![Page 16: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/16.jpg)
• translates horizontally units.
![Page 17: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/17.jpg)
• translates horizontally units.
![Page 18: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/18.jpg)
• translates horizontally units.
![Page 19: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/19.jpg)
Examples
Find the equation of the relation under the translation vectorindicated. Graph both the original and translated relations on the same set of axes.
(a) (b)
![Page 20: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/20.jpg)
Example (a)
![Page 21: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/21.jpg)
Example (a)
![Page 22: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/22.jpg)
Example (b)
![Page 23: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/23.jpg)
Example (b)
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Summary
![Page 25: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/25.jpg)
• translates by the vector
EXAMPLE:
Find the equation of under the translation
![Page 26: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/26.jpg)
Find the equation of under the translation
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Dilation from the x-axis
• is a vertical stretch of with dilation factor .
![Page 28: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/28.jpg)
Dilation from the x-axis
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Dilation from the x-axis
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Dilation from the x-axis
![Page 31: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/31.jpg)
Dilation from the x-axis
![Page 32: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/32.jpg)
Dilation from the x-axis
![Page 33: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/33.jpg)
Dilation from the y-axis
• is a horizontal stretch of with dilation factor .
![Page 34: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/34.jpg)
Dilation from the y-axis
![Page 35: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/35.jpg)
Dilation from the y-axis
![Page 36: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/36.jpg)
Dilation from the y-axis
![Page 37: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/37.jpg)
Dilation from the y-axis
![Page 38: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/38.jpg)
Dilation from the y-axis
![Page 39: Functions and equations](https://reader035.vdocuments.us/reader035/viewer/2022062301/56816109550346895dd054e7/html5/thumbnails/39.jpg)
Reflections
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Reflection about the x-axis
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Reflection about the y-axis