Warm-up: Graph y =|x| y = x2 y = HW: p. 168-171 (5 skip e, 7-11 all, 15, 19-25 odd, 29, 33, 35, 56, 57, 62, 63, 64)

Objective: Shift, reflect, and stretch graphs combinations of functions composition of functions

Parent Functions:

Translations – In General The transformation of any function f(x) looks like: a represents the vertical stretch and orientation h represents the horizontal shift k represents the vertical shift.

Use the graph of f (x) = x3 to graph g (x) = (x – 2)3 and h(x) = x3 + 4 .

Shifting Graphs Graph the function. down 2

Shifting Graphs Graph the function. left 3 down 4

Shifting Graphs Graph the function. right 2 down 5

Reflection About the x-Axis Y=f(x) Y=-f(x)

Reflect over the x-axis Reflecting Graphs Graph the function. Reflect over the x-axis

Reflect over the x-axis Reflecting Graphs Graph the function. left 1 Reflect over the x-axis up 2

Reflection About the y-Axis Y=f(-x) Y=f(x)

Reflect over the y-axis Reflecting Graphs Graph the function. Reflect over the y-axis

Vertical Stretching and Shrinking Graphs

Vertical Stretching and Shrinking Graphs Y=f(x) Y=2f(x) Y=1/2f(x)

three times the y-value Nonrigid Transformations Graph the function. three times the y-value

left 1 down 2 one fourth the y-value Nonrigid Transformations Graph the function. left 1 down 2 one fourth the y-value

Combine like terms & put in descending order The sum f + g Combine like terms & put in descending order

The difference f - g Distribute negative CAUTION: Make sure you distribute the – to each term of the second function Distribute negative

Should put in descending order. The product f • g FOIL Should put in descending order.

Nothing more you could do here. (If you can reduce these you should). The quotient f /g Nothing more you could do here. (If you can reduce these you should).

FOIL first and then distribute the 2 The Composition Function This is read “f composition g” FOIL first and then distribute the 2

You could multiply this out but since it’s to the 3rd power we won’t This is read “g composition f” You could multiply this out but since it’s to the 3rd power we won’t

This is read “f composition f”

The DOMAIN of the Composition Function The domain of f composition g is the set of all numbers x in the domain of g such that g(x) is in the domain of f. The domain of g is x  1

Summary: Shift, reflect, and stretch graphs combinations of functions composition of functions

  HW: p. 168-171 (5 skip e, 7-11 all, 15, 19-25 odd, 29, 33, 35, 56, 57, 62, 63, 64)