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.

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.

Translations y = ƒ(x - a) + b

The role of parameter a:a > 0 the graph shifts right a units.

- the x-coordinates are increased a units.

a < 0 the graph shifts left a units.- the x-coordinates are decreased a units.

WARNING: watch the sign of a

The role of parameter b:b > 0 the graph shifts up b units.

- the y-coordinates are increased b units.

b < 0 the graph shifts down b units.- the y-coordinates are decreased b units.

Examples

DICTIONARY

Stretches ... (and a wee bit about reflections)

Let's start with a circle ...

Let's look at some graphs ... We'll head over to fooplot.com ...

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

DICTIONARY

Putting it all together ...

y = ƒ(x)

REMEMBER: stretches before translations

Try these examples ...

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 DICTIONARY

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...

DICTIONARY

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:

DICTIONARY