PARENT FUNCTIONS & TRANSFORMATIONS

Parent functions are common functions of algebra that we .

A is when we , or a graph.

The graph maintains the same general characteristics, but it morphed.

transform

transformationshift reflect

stretch/compress

Graph the parent function using a table of values

1.

y = x

Graph the parent function using a table of values

2.

y =x2

Graph the parent function using a table of values

3.

y =x3

Graph the parent function using a table of values

4.

y = x

A is a slide. When we graph translations, we will the graph from one location to another.

translationshift

Vertical Shifts

A shift moves the graph or .

If is our function, the graph of is the graph of shifted c units vertically upward.

If is our function, the graph of is the graph of shifted c units vertically downward.

Note: with vertical shifts, the sign on the movement (+/-) is the direction as what the shift is. So, if the sign is positive we go and when the sign is negative we go .

Vertical shifts are never the parentheses.

f x( )

f x( )

f x( )

f x( )

y = f x( ) + c

y = f x( )−c

vertical updown

sameup

downinside

When looking at the equation of a parent function, a vertical shift is grouped with the x.

not

Graph the function

5.

y = x+ 3

Graph the function

6.

y =x3 −5

Horizontal Shifts

A shift moves the graph or .

If is our function, the graph of is the graph of shifted c units horizontally to the left.

If is our function, the graph of is the graph of shifted c units horizontally to the right.

Note: with horizontal shifts, the sign on the movement (+/-) is the direction as what the shift is. So, if the sign is positive we go

and when the sign is negative we go .

Horizontal shifts are inside the parentheses.

f x( )

f x( )

y = f x+ c( )

f x( )

f x( )

y = f x−c( )

horizontal leftright

oppositeleft

rightalways

Graph the function

7.

y = x−3( )

2

Graph the function

8.

y = x+ 2

Vertical Reflections

A reflection is a reflection about the .

The graph of is the graph of reflected about the x-axis.

In other words, if there is a out front, we flip over the .

y =−f x( )

y = f x( )

vertical x-axis

negativex-axis

Graph the function

9.

y =−x3

Graph the function

10.

y =− x

Combining Shifts and Reflections

We can graph both a vertical and horizontal shift at the same time.

Just remember the vertical movement is the parentheses, and the horizontal movement is the parentheses.

A vertical reflection will also be the parentheses.

Order of transformations – work from the inside out! Horizontal Shift Vertical Reflection Vertical Shift

outside

insideoutside

Graph the function

11.

y = x−6( )

3−1

Graph the function

12.

y = x−1+ 4

Graph the function

13.

y =− x+5( )

2−2

Graph the function

14.

y =−x+ 3 + 6