1.1: Basic Functions and Translationsdobsonmath.weebly.com/uploads/1/1/8/0/11809374/pc_12_ch.1_notes.pdf1.1: Basic Functions and Translations ... ( 2) 12 2. Radical functions ... Function

Pre-calculus 12 Chapter 1: Function Transformations

1

1.1: Basic Functions and Translations

Here are the Basic Functions (and their coordinates!) you need to get familiar with.

1. Quadratic functions (a.k.a. parabolas)

2y x

Ex. 2( 2) 1y x

2. Radical functions (a.k.a. square root function)

y x

Ex. 3 4y x

3. Absolute-value functions

| |y x

Ex. | 1| 2y x

4. Reciprocal functions

1

yx

Ex. 1

12

yx

Note: You are expected to remember the shape of the above (left) functions!


2

Regardless of the type of function ( )y f x , the transformed function

y f (xh) k tells us:

Ex. 1: Find the equations for the base functions and their transformed graphs.

a)

Base function: Transformed function:

b)


c)


“h” = ________________________________

“k”= ________________________________


3

Ex. 2: For the function

y4 f (x2) state the value of h and k that represent the horizontal and vertical

translations applied to

y f (x)

Ex. 3: Determine the new function when ( 6) 1y f x is translated 4 units to the left and 2 units

downward.

Ex. 4: Transform the following graph. Describe the transformations in words.

Given: ( )y f x

Graph: ( 2) 3y f x

Transformation: __________________________________________________________

**To translate, choose key points on the graph and then translate each one to graph its corresponding image

point on the transformed graph.


4

1.2 (part 1): Vertical and Horizontal Reflections

A reflection can be identified with a “negative sign.” A reflection is a mirror image of a given function.

Using a graphing calculator, let’s explore the effect of having a “negative sign” at different locations of a

function.

Ex. 1: Graph 2y x and 2y x on the same grid.

Ex. 2: Graph | |y x and | |y x on the same grid.

Ex. 3: Graph y x and y x on the same grid.


5

Observations:

y f (x) ______________________________________________________ Mapping: ( , )

y f (x) ______________________________________________________ Mapping: ( , )

Ex. 4: Without using a graphing calculator,

a) graph ( ) 3 2f x x with a solid line.

____ b) graph ( )y f x with a dotted line.

. . . c) graph ( )y f x with a broken line.

_ _ _

Ex. 5: Given 3 24 2 1f x x x , write a new function after applying the following reflection:

a) over the x-axis b) over the y-axis


6

Ex. 6: Given

f (x), graph the indicated relation. State the domain and

range for each of them. Determine if it is a function.

a) Graph y f x

Domain: Range: Function?

b) Graph y f x

Domain: Range: Function?

Homework: p. 28 – 29: # 1, 3, 4, 5c , 5d, 7b, 7d

* use graph paper when drawing graphs (available from Ms. Dobson)

* “mapping notation”: (x,y) ( , )


7

1.2 (part 2): Expansions and Compressions

Ex. 1: Consider the following function:

f x x

Graph the indicated function using the table of values on the graphing calculator:

a) 2y f x

b) 1

2y f x

c) 2y f x

d) 1

2y f x


8

Observations:

y af (x) ______________________________________________________ Mapping: ( , )

y f (bx) ______________________________________________________ Mapping: ( , )

Ex. 2: Using )(xfy as a base function, describe the transformation when x is replaced by 2x and y is

replaced by y3

1 in words and in mapping notation.

Ex. 3: Given the graph of

f (x), perform each of the following transformations:

a) a vertical expansion by a factor of 2 b) a horizontal compression by a factor of

1

2

Homework: p. 28 – 31: # 2, 5a, 5b, 6, 7a, 7c, 8, 9, 14

* use graph paper when drawing graphs (available from Ms. Dobson)


9

1.3 (part 1): Combining Transformations

Does order matter? Let’s explore.

Ex. 1: Graph y x

Vertical Expansion first vs. Vertical Translation first: (Reverse Order)

1) VE by factor of 2 1) VT by 1 unit down

2) VT by 1 unit down 2) VE by a factor of 2

Did the order effect the outcome?

Horizontal Expansion first vs. Horizontal Translation first: (Reverse Order)

1) HE by factor of 2 1) HT by 1 unit right

2) HT by 1 unit right 2) HE by a factor of 2

Did the order effect the outcome?

x y


10

“Build-it-up” Method: Replace x with “new x” – Put in “new y” setting.

Ex. 2: Given the description, write the following transformations in function notation.

a) VE of 2

VT down 3

HT right 1

HC of 1/3

b) HE of 2

VC of 1/6

VT up 1

HT left 6

Combinations of Transformations:

We will perform transformations in the order S (Stretches), R (Reflections), T (Transformations)

First, re-write the function as

y af (b(xh)) k to be able to read all of the transformations directly.

Notice: there is no coefficient on x it must be factored out!

We will perform the transformations in order SRT for both vertical and horizontal:

1) A Vertical Stretch by a factor of a; a Horizontal Stretch by a factor of

1

b

2) A Vertical Reflection if a < 0 in the x-axis; a Horizontal Reflection if b < 0 in the y-axis

3) A Vertical Translation by a factor of k; a Horizontal Translation by a factor of –h

Ex. 3: Describe the order of transformations that occur for the following functions.

a) 3 2 4y f x b) 2 1y f x

c) 2 1y f x d) 1

2 1 53

y f x


11

Ex. 4: Given ( )y f x :

a) Describe the transformation 2 1 2 3y f x

b) Graph the indicated functions on the grid provided.

Ex. 5: Write the equation for both the base function and the

transformed function.

Math 12 Pre-Calculus Chapter 1: Function Transformations

12

1.4: Inverse Functions The inverse of a relation is found by interchanging the x and y coordinates of the ordered pairs of that relation. Mapping: (x,y) ( , ) Graphically, this is the same as a reflection across the line y = x

The notation for an inverse is )(1 xf

Ex. 1: Given the function 32)( xxf :

a) Determine )(1 xf

b) Graph )(xf and )(1 xf on the same grid

c) Invariant point(s):

Ex. 2: Consider the function 4)( 2 xxf

a) Graph f(x) on the grid provided and state the

domain and range.

b) Find the inverse of the function 4)( 2 xxf .

c) Graph the inverse and state the domain and range.

d) Is the inverse of f a function? If not, how could the domain or range of f(x) be restricted so

that the inverse of f is a function?

Recall: an invariant point is any point that is unchanged

Math 12 Pre-Calculus Chapter 1: Function Transformations

13

Ex. 3: For each of the following functions

Find the inverse function using the notation )(1 xf , where appropriate

State the domain and range of the given function and its inverse.

a) 2)( xxf b) 2)3()( xxf c) 7

( )3

xf x

x

Ex. 4: a) Accurately graph the inverses of the following functions on the same grid.

b) Is the inverse a function? Justify your answer.

Homework: p. 51 – 53: # 2, 4ac, 5ace, 7a, 9ce, 11, 13 (use graphing calculator), 15

8

6

4

2

-2

-4

-6

-8

-10 -5 5 10

8

6

4

2

-2

-4

-6

-8

-10 -5 5 10

Documents

1.1: Basic Functions and Translationsdobsonmath.weebly.com/uploads/1/1/8/0/11809374/pc_12_ch.1_notes.pdf1.1: Basic Functions and Translations ... ( 2) 12 2. Radical functions ... Function