reflections, transformations and interval notation l. waihman revised 2006
TRANSCRIPT
![Page 1: Reflections, Transformations and Interval Notation L. Waihman Revised 2006](https://reader036.vdocuments.us/reader036/viewer/2022082820/56649e975503460f94b9a931/html5/thumbnails/1.jpg)
Reflections, Transformations and Interval Notation
L. Waihman Revised 2006
![Page 2: Reflections, Transformations and Interval Notation L. Waihman Revised 2006](https://reader036.vdocuments.us/reader036/viewer/2022082820/56649e975503460f94b9a931/html5/thumbnails/2.jpg)
Reflections of
The graph of is the reflection of the graph of over the x-axis.The graph of is the reflection of the graph of over the y-axis.The graph of is the reflection of the graph of over the line y=x.
y=f x
y f x
y f x
1y f x
f
f
f
![Page 3: Reflections, Transformations and Interval Notation L. Waihman Revised 2006](https://reader036.vdocuments.us/reader036/viewer/2022082820/56649e975503460f94b9a931/html5/thumbnails/3.jpg)
Identify the reflection in each of the graphs below.
g
x
x
f x
x
g
x
x
f x
x
33 6
2
g
xx
x
f
x
@ x-axis @ y-axis @ line y=x
![Page 4: Reflections, Transformations and Interval Notation L. Waihman Revised 2006](https://reader036.vdocuments.us/reader036/viewer/2022082820/56649e975503460f94b9a931/html5/thumbnails/4.jpg)
Transformations
If is negative, the graph is reflected over the x-axis. determines the vertical stretch or shrink. If >1, the graph has a vertical stretch by a factor of . If
the graph has a vertical shrink by a factor of .
y a b x c d function
a
aa
a 0 1,a
a
![Page 5: Reflections, Transformations and Interval Notation L. Waihman Revised 2006](https://reader036.vdocuments.us/reader036/viewer/2022082820/56649e975503460f94b9a931/html5/thumbnails/5.jpg)
More Transformations is used to determine the horizontal stretch or shrink of the graph. If , the graph shrinks horizontally
units. If , the graph stretches
horizontally b units.If b is negative, the graph reflects about the y-axis.
b
1b 1b
0 1b
![Page 6: Reflections, Transformations and Interval Notation L. Waihman Revised 2006](https://reader036.vdocuments.us/reader036/viewer/2022082820/56649e975503460f94b9a931/html5/thumbnails/6.jpg)
More Transformations
determines the horizontal slide of the graph. If the slide is to the right. If
the slide is to the left.
determines the vertical slide of the graph. If the slide is down. If the slide is up.
cc 0,
c 0,d
0,d 0,d
![Page 7: Reflections, Transformations and Interval Notation L. Waihman Revised 2006](https://reader036.vdocuments.us/reader036/viewer/2022082820/56649e975503460f94b9a931/html5/thumbnails/7.jpg)
More Transformations
There is an order of operations to be followed when applying multiple transformations.This order of transformations is BCAD.
![Page 8: Reflections, Transformations and Interval Notation L. Waihman Revised 2006](https://reader036.vdocuments.us/reader036/viewer/2022082820/56649e975503460f94b9a931/html5/thumbnails/8.jpg)
Example:
parent function: quadratic or squaring function
Stretch: Horizontal shrink by
Slide: left 2 unit
Reflection: over the x-axis
Stretch: vertical stretch by a factor of 3
Slide: up 1 unit
y x2
3 2 4 1
12
![Page 9: Reflections, Transformations and Interval Notation L. Waihman Revised 2006](https://reader036.vdocuments.us/reader036/viewer/2022082820/56649e975503460f94b9a931/html5/thumbnails/9.jpg)
Absolute Value TransformationsAbsolute Value of the Function indicates that all the negative y-values must become positive so you need to reflect any parts normally below the x-axis above the x-axis. Absolute Value of the Argument indicates all negative x-values would produce the same results as the positive counterparts, thus everything to the left of the y-axis would disappear and the remaining graph to the right of the y-axis would reflect across the
y-axis.
![Page 10: Reflections, Transformations and Interval Notation L. Waihman Revised 2006](https://reader036.vdocuments.us/reader036/viewer/2022082820/56649e975503460f94b9a931/html5/thumbnails/10.jpg)
Absolute Value of the function:
![Page 11: Reflections, Transformations and Interval Notation L. Waihman Revised 2006](https://reader036.vdocuments.us/reader036/viewer/2022082820/56649e975503460f94b9a931/html5/thumbnails/11.jpg)
Absolute Value of the Argument:
![Page 12: Reflections, Transformations and Interval Notation L. Waihman Revised 2006](https://reader036.vdocuments.us/reader036/viewer/2022082820/56649e975503460f94b9a931/html5/thumbnails/12.jpg)
Absolute Value of the Function and the Argument
![Page 13: Reflections, Transformations and Interval Notation L. Waihman Revised 2006](https://reader036.vdocuments.us/reader036/viewer/2022082820/56649e975503460f94b9a931/html5/thumbnails/13.jpg)
Interval Notation From this point forward we will be using interval notation instead of set notation to describe solutions.Interval notation is written with either parenthesis or brackets and contains the first element of the interval, the second element.Parenthesis are used if the element is NOT included in the solution. Brackets are used if the element is included in the solution.
![Page 14: Reflections, Transformations and Interval Notation L. Waihman Revised 2006](https://reader036.vdocuments.us/reader036/viewer/2022082820/56649e975503460f94b9a931/html5/thumbnails/14.jpg)
If the domain of a function is , we now describe this as extending from negative infinity to positive infinity – written .
Note that infinity cannot be found so it cannot be an included element.If the domain consisted of the x-values from 0 to infinity with 0 being on the graph, we would write this as .
,
0,
![Page 15: Reflections, Transformations and Interval Notation L. Waihman Revised 2006](https://reader036.vdocuments.us/reader036/viewer/2022082820/56649e975503460f94b9a931/html5/thumbnails/15.jpg)
Interval Notation
If the range is from -3 to infinity with a y-value at -3, we would write the range as .If the range contained the y-values from 0 to 6 inclusive, we would write the solution as .
3,
0,6
![Page 16: Reflections, Transformations and Interval Notation L. Waihman Revised 2006](https://reader036.vdocuments.us/reader036/viewer/2022082820/56649e975503460f94b9a931/html5/thumbnails/16.jpg)
Put it all together!Graph the following function:
List all the transformations including reflections.Determine the symmetry of the function.Determine if the function is even, odd or neither.Identify the domain and range using interval notation.
23(6 2 ) 4 f x x