pre-calculus lesson 3: translations of function graphs vertical and horizontal shifts in graphs of...

Pre-Calculus Lesson 3: Translations of Function GraphsVertical and horizontal shifts in graphs of various functions

The Absolute Value FunctionWhat does the graph of y = |x| look like?

y =|x|

y

x

10

10

-10-10

y =|x|+3

y

x

10

10

-10-10

y =|x|+3

y

x

10

10

-10-10

y =|x|-3

y

x

10

10

-10-10

y =|x|-3

y

x

10

10

-10-10

Observation #1:Adding to or subtracting from a function will shift the graph of the parent function up or down, respectively.

y =|x+3|

y

x

10

10

-10-10

y =|x+3|

y

x

10

10

-10-10

y =|x-3|

y

x

10

10

-10-10

y =|x-3|

y

x

10

10

-10-10

Observation #2:Adding to or subtracting from the variable will shift the graph of the parent function left or right, respectively.

y =-|x|

y

x

10

10

-10-10

y =-|x|

y

x

10

10

-10-10

http://www.uiowa.edu/~examserv/mathmatters/tutorial_quiz/geometry/typesofsymmetry_clip_image017.gif

f(x)f(-x)

Observation #3:Negating the function will reflect the parent function over the x-axis. Negating the variable will reflect the parent function over the y-axis.

Even and Odd Functions An even function

is: Symmetric with

respect to the ______________.

Defined by the equation ______________.

An odd function is: Symmetric with

respect to the ______________.

Defined by the equation ______________.

Even and Odd Functions An even function

is: Symmetric with

respect to the ______________.

Defined by the equation ______________.

An odd function is: Symmetric with

respect to the ______________.

Defined by the equation ______________.

y-axis

Even and Odd Functions An even function

is: Symmetric with

respect to the ______________.

Defined by the equation ______________.

An odd function is: Symmetric with

respect to the ______________.

Defined by the equation ______________.

y-axis

)()( xfxf

Even and Odd Functions An even function

is: Symmetric with

respect to the ______________.

Defined by the equation ______________.

An odd function is: Symmetric with

respect to the ______________.

Defined by the equation ______________.

y-axis origin

)()( xfxf

Even and Odd Functions An even function

is: Symmetric with

respect to the ______________.

Defined by the equation ______________.

An odd function is: Symmetric with

respect to the ______________.

Defined by the equation ______________.

y-axis origin

)()( xfxf )()( xfxf

Determine whether each function is even, odd, or neither.

64 xxxf

Determine whether each function is even, odd, or neither.

64 xxxf

648333 64 f

Determine whether each function is even, odd, or neither.

64 xxxf

648333 64 f

648333 64 f

Determine whether each function is even, odd, or neither.

64 xxxf

648333 64 f

648333 64 f

Determine whether each function is even, odd, or neither.

52 33 xxxf

Determine whether each function is even, odd, or neither.

52 33 xxxf

10823232 52 f

Determine whether each function is even, odd, or neither.

52 33 xxxf

10823232 52 f

8423232 52 f

Determine whether each function is even, odd, or neither.

52 33 xxxf

10823232 52 f

8423232 52 f

Determine whether each function is even, odd, or neither.

xxxf 3

Determine whether each function is even, odd, or neither.

xxxf 3

130555 3 f

Determine whether each function is even, odd, or neither.

xxxf 3

130555 3 f

130555 3 f

Determine whether each function is even, odd, or neither.

xxxf 3

130555 3 f

130555 3 f

PracticeDescribe the shifts in the graph of the function from the parent function of . State the coordinates of the vertex point of the graph. Determine if the function is odd, even, or neither and justify your answer.

13 2 xxf 2xxf