A. Function Transformers

Pre-Calculus 30

PC30.7 Extend understanding of transformations to

include functions (given in equation or graph form) in general, including horizontal and vertical translations, and horizontal and vertical stretches.

PC30.8 Demonstrate understanding of functions,

relations, inverses and their related equations resulting from reflections through the: x-axis, y-axis, line y=x

Key Terms Transformations Mapping Translations Image Point Reflection Invariant Point Stretch Inverse of a Function Horizontal Line Test

1. Translations PC30.7 Extend understanding of transformations to

include functions (given in equation or graph form) in general, including horizontal and vertical translations, and horizontal and vertical stretches.

1. Translations First off, a transformation is when a functions

equation is altered resulting in any combination of location, shape and/or orientation changes of the graph

Every point on the original graph corresponds to a point on the transformed graph

The relationship between the points is called mapping

Mapping Notation is a way to show the relation between the original function and the transformed function.

original (x,y) translation (x,y+3)

Mapping Notation: (x,y) (x,y+3)

Translation is a type of transformation

A translation can move a graph left, right, up and down.

In a translation the location of the graph changes but not the shape or orientation.

Lets look at a quick example to see how a translation works and what it looks like in an equation

Graph: y=x2 , y-2=x2 , y=(x-5)2

Now before we graph the following 3 functions let’s predict what we think will happen?

Graph: y=x2 , y+1=x2 , y=(x+3)2

So with vertical and horizontal translations we shift the graph of a function vertically and/or horizontally by applying one or both of the changes to the equation

Vertical Shift: y-k=f(x)

Horizontal Shift: y=f(x-h)

Both: y-k=f(x-h)

Example 1

Sketch a graph of

Example 2

Key Ideas p. 12

Practice Ex. 1.1 (p.12) #1-14

#1-13 odds, 17-19

2. Reflect and Stretch PC30.7 Extend understanding of transformations to

include functions (given in equation or graph form) in general, including horizontal and vertical translations, and horizontal and vertical stretches.

PC30.8 Demonstrate understanding of functions,

relations, inverses and their related equations resulting from reflections through the: x-axis, y-axis, line y=x

2. Reflect and Stretch A Reflections of a functions graph is the mirror

image in a line called the Line of Reflection

Reflections do not change the shape of the graph but does change the orientation of the graph

When output of a function is multiplied by -1 the result is y=-f(x)

Vertical Reflection (reflect in x-axis) (x,y)(x,-y) Line of reflection=x-axis

When input of a function is multiplied by -1 the result is y=f(-x)

Horizontal Reflection (reflect in y-axis) (x,y)(-x,y) Line of reflection=y-axis

Example 1

A Stretch changes the shape of a graph but not its location

A vertical stretch can make the function shorter or taller bc the stretch multiplies or divides the y-values by a constant while the x is unchanged

Shorter (vert compression)

(x,y)(x, y) Use IaI because the

negative is used in reflection

Taller (vert expansion)

(x,y)(x, ay) Use IaI because the

negative is used in reflection

A Horizontal Stretch can make the function narrower or wider because the stretch multiplies or divides the x-values by a constant while the y-values are unchanged

Narrower (horiz compression)

(x,y)(x, y) Use IbI because the

negative is used in reflection

Wider (horiz expansion)

(x,y)(bx, y) Use IbI because the

negative is used in reflection

If the a or b values are negative there would also be a reflection.

Example 2

Example 3

a) b)

Example 4

Key Ideas p. 27

Practice Ex. 1.2 (p.28) #1-12

#1-6, 7-9 odds in each, 10-12, 15, 16

3. Combining Transformations

PC30.7 Extend understanding of transformations to

include functions (given in equation or graph form) in general, including horizontal and vertical translations, and horizontal and vertical stretches.

PC30.8 Demonstrate understanding of functions,

relations, inverses and their related equations resulting from reflections through the: x-axis, y-axis, line y=x

3. Combining Transformations

Multiple transformations can be applied to a function using the General Transformation Model

y-k=af(b(x-h)) or y=af(b(x-h)) +k

The same order of operations are used as when you are working with numbers (BEDMAS)

So multiplying and dividing (stretches, reflections) are done first then add and subtract (translations)

Steps to graph combinations:1. Horizontal stretch and reflect in the y-axis (if b<0)2. Vertical stretch and reflect in the x-axis (if a<0)3. Horizontal and/or vertical Translations (h and k)

Lets look at the transformations in mapping notation for y=af(b(x-h)) +k

Example 1

Example 2

Example 3

Key Ideas p.38

Practice Ex. 1.3 (p.38) #1-12 odds in each with multiple

parts #3-16 odds in each with multiple

parts

4. Inverse Functions PC30.8 Demonstrate understanding of functions,

relations, inverses and their related equations resulting from reflections through the: x-axis, y-axis, line y=x

4. Inverse Functions

The Inverse of a Function y=f(x) is denoted y=f -

1(x) if the inverse is a function.

The -1 is not an exponent because f represents a function, not a variable. (just like in sin -1(x))

The inverse of a function reverses the processes represented by that function.

For example, the process of squaring a number is reversed by taking the square root. Taking the reciprocal of a number is reversed by taking the reciprocal again.

For example, for f(x)=2x+1 we are multiplying by 2 and adding 1.

What would the inverse be?

To determine the inverse of a function, interchange the x and y coordinates

Function Inverse(x,y) (y,x)y=f(x) x=f(y)reflect in the line y=x

When working with an equation of a function y=f(x), interchange the x for y.

Then solve for y to get the equation for the inverse, if the inverse is a function, then y=f -1(x)

If the inverse of a function is not a function (recall the vertical line test), restrict the domain of the base function so that the inverse becomes a function.

You will see the frequently with quadratic functions.

For example, the inverse of f(x)=x2, x≥0, is f -

1(x)= . The inverse is a function only if the domain of the base function is restricted.

Restricting the domain is necessary for any function that changes direction (increasing to decreasing, or vise versa) at some point in the domain of the function

Unrestricted domain Restricted domain x≤0

Example 1

Example 2

Example 3

Key Ideas p. 51

Practice Ex. 1.4 (p.51) #1-16 odds in questions with

multiple parts #4-20 odds in questions with multiple

parts