The document discusses transformations of graphs including: 1) Symmetry with respect to the x-axis, y-axis, and origin. 2) Translations that shift graphs horizontally or vertically by adding/subtracting constants. 3) Reflections that flip graphs across the x-axis or y-axis. 4) Stretching/compressing graphs vertically or horizontally by multiplying the function by constants greater than or less than 1.

1. 1

2. 2
I) Symmetry „Symmetry with Respect to the x-axis:If (x,y) on the graph, then (x, -y) is also in the graph „Symmetry with Respect to the y-axis:If (x,y) on the graph, then (-x, y) is also in the graph „Symmetry with Respect to the origin:If (x,y) on the graph, then (-x, -y) is also in the graph

3. 3
xxyyxxHWyxxyxy= −= =− = += )5)()46)34)21)1222Ex1:Determine whether the graph of each equation is symmetric with respect to the a. x-axis, b. y-axis, c. origin
Ex2:Find the image of the point R(-2, 3) with respect to the x-axis, y-axis, and the origin

4. 4
II) Even and Odd Functions
•A function f is evenif f(-x) = f(x) for every xin the domain of f.
•A function fis oddif f(-x) = -f(x) for every xin the domain of f.
Ex3:Determine each function is even, odd , or neither. 534232332)()61)()55)()4)()3)()2)()1xxxxfxxxfxkxxxhxxgxxf− = + = =+= ==

5. 5
Notes: xyf(x)= x3 (odd) xyf(x)=x2(even) f (x)=x2–xneither* If the graph is not symmetric with respect to the y-axis or the origin, then function is neither even nor odd*The graph of an even function is symmetric with respect to they-axis*The graph of an odd function is symmetric with respect to with respect to the origin

6. 6
f(x) f(x) + c+cf(x) –c-c
•If cis a positive real number, the graph of f (x) + cis the graph of y= f (x)shifted upwardcunits.
Vertical Translations•If cis a positive real number, the graph of f (x) –cis the graph of y= f(x)shifted downwardcunits. xyIII) Translations of Graphs

7. 7
xyy = f(x)y = f(x –c) +cy = f(x + c) -c
•If cis a positive real number, then the graph of
f (x –c) is the graph of y= f (x)shifted to the rightcunits.
Horizontal Translations
•If cis a positive real
number, then the graph
of f (x + c) is the
graph of y= f (x)
shifted to the left
cunits.

8. 8
2)f(x)=|x| xy1)y = x 2xyxy3)f (x)= x3-4xy4xy=)4Famous Graphs:

9. 9
Ex4:Sketch the graph of the following functions
1) y=x2
2) y=x2+1
3) y=x2-1
4)y=(x+1)2
5) y=(x-1)2
6)y=(x-1)2+2
7) y=(x+1)2+2

10. 10
Ex6:If the graph of the function 31+ − = xxyis shiftedhorizontally two units to the left and vertically Three units up, hen find the equation of the new graph. Ex5:Find the range of the function 31)(−+=xxf

11. 11
•The graph of the function
y =–f (x) is the graph of
y =f (x)reflected in the
x-axis. IV) Reflections of Graphs
•The graph of the function y =f (–x)is the graphof y=f x) reflected in the y-axis. xyy =f(–x)y =f(x) y =–f(x)

12. 12
Ex7:Sketch the graph of the following functions: 21)921)821)71)61)51)41)3)2)1++−−= −+−= +−−= +−= +−= +−= +−= −= −= xyxyxyxyxyxyxyxyxy

13. 13
Vertical Stretching and Compressing
•If c> 1then the graph of y= c f(x)is the graph of y= f(x) stretchedvertically by c.
•If 0 < c< 1then the graph of y= c f(x)is the graph of y= f(x) shrunkvertically by c. –4xy44y =x2is the graphof y =x2compressed vertically by. 241xy= 41241xy= y =2x2V) Stretching and Compressing of Graphsy=2x2 is the graph of y =x2stretched verticallyby 2. Example:

14. 14
-4xy44y = |x| y = |2x|
Horizontal Compressing and Stretching
•If c> 1, the graph of y= f(cx)is the graph of y= f(x)
compressed horizontally by c.
•If 0 < c< 1, the graph of y= f(cx)is the graph of y= f(x) stretchedhorizontallyby c. y =|2x|is the graph of y =|x| shrunk horizontally by 2.xy21= is the graph of y =|x| stretched horizontallyby . xy21= 21Example:

15. 15
-4 448xyEx8:Graph using the graph ofy =x3.3)1( 213++=xy3)1( 213++=xyStep 4: -4 448xy
Step 1: y =x3
Step 2: y =(x + 1)33)1( 21+=xyStep 3: Graph y =x3and do one transformation at a time.

16. 16
Ex:If the point (-1, 2) lies on the graph of y=f(x), then find the image of this point on the graph of the following functions: 2)12(3))(51)(3)4) 31(2)3)2()2)(2)1+−−= += = = = xfyHWxfyxfyxfyxfy