pre-cal 40s slides february 29, 2008
DESCRIPTION
Translations, reflections, even and odd functions. Introduction to inverses.TRANSCRIPT
Stretches and Compressions:
The role of parameter a:a > 1 the graph of ƒ(x) is stretched vertically.0 < |a| < 1 the graph of ƒ(x) is compressed vertically.- the y-coordinates of ƒ are multiplied by a.
The role of parameter b:b > 1 the graph of ƒ(x) is compressed horizontally.(Everything "speeds up")0<|b|<1 the graph of ƒ(x) is stretched horizontally. (Everything "slows down")- the x-coordinates are multiplied by .
Examples
Putting it all together ...
y = ƒ(x)
REMEMBER: stretches before translations
Try these examples ...
Practice what you've learned ...
The coordinates of a point, A, on the graph of y = ƒ(x) are (-2, -3). What are the coordinates of it's image on each of the following graphs:
The image of point B after each transformation shown above is given below as point C(n). Find the original coordinates of B.
C1 (2, 3) C3 (5, -4)C2 (-3, 7) C4 (-1, 6) C5 (-4, -2)
Consider the equation below. Which transformation do you think should be applied first? second? third? fourth?
Given A(-2, -3) find the coordinates of its image under the transformation given above.
The image of point B after the transformation shown above is (1, 4). Find the original coordinates of B.
Vertical ReflectionsGiven any function ƒ(x):-ƒ(x) produces a reflection in the x-axis.y-coordinates are multiplied by (-1)
WARNING:
undoes whatever ƒ did.
Inverses: the inverse of any function ƒ(x) is (read as: "EFF INVERSE")
Horizontal ReflectionsGiven any function ƒ(x):ƒ(-x) produces a reflection in the y-axis.x-coordinates are multiplied by (-1)
Reflections
EVEN FUNCTIONSGraphically: A function is "even" if its graph is symmetrical about the y-axis.
These are not ...
Examples: Are these functions even?
1. f(x) = x² 2. g(x) = x² + 2x f(-x) = (-x)² g(-x) = (-x)² + 2(-x) f(-x) = x² g(-x) = x² - 2xsince f(-x)=f(x) since g(-x) is not equal to g(x)f is an even function g is not an even function
Symbolically (Algebraically)a function is "even" IFF (if and only if) ƒ(-x) = ƒ(x)
These functions are even...
ODD FUNCTIONSGraphically: A function is "odd" if its graph is symmetrical about the origin.
These are not ...
1. ƒ(x) = x³ - x 2. g(x) = x³- x² ƒ(-x) = (-x)³ - (-x) g(-x) = (-x)³ - (-x)² ƒ(x) = -x³ + x g(x) = -x³ - x²
-ƒ(x) = -(x³ - x) -g(x) = -(x³-x²)-ƒ(x) = -x³ + x -g(x) = -x³+ x²
since ƒ(-x)= -ƒ(x) since g(-x) is not equal to -g(x)ƒ is an odd function g is not an odd function
These functions are odd ...
Symbolically (Algebraically)a function is "odd" IFF (if and only if) ƒ(-x) = -ƒ(x)
Examples: