

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









Transformations
Mapping
Translations
Image Point
Reflection
Invariant Point
Stretch
Inverse of a Function
Horizontal Line Test


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.

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)

Mapping Notation:
translation (x,y+3)
(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:

Horizontal Shift: y=f(x-h)

Both:
y-k=f(x)
y-k=f(x-h)
Sketch a graph of 𝑦 = (𝑥 + 5)2 −2

Ex. 1.1 (p.12) #1-14
#1-13 odds, 17-19




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

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

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)
𝑦
=
1
𝑓
𝑎
𝑥
 (x,y)(x,
 Use
1
y)
𝑎
IaI because the
negative is used in
reflection
 Taller
(vert expansion)
 𝑦 = 𝑎𝑓 𝑥
 (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)
 𝑦 = 𝑓 𝑏𝑥
1
 (x,y)(
𝑏
 Use
x, y)
IbI because the
negative is used in
reflection
 Wider
𝑦
=𝑓
(horiz expansion)
1
𝑥
𝑏
 (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.
a) 𝑔 𝑥 = 𝑓(3𝑥)
b) 𝑔 𝑥 =
1
𝑓( 𝑥)
4

Ex. 1.2 (p.28) #1-12
#1-6, 7-9 odds in each, 10-12, 15, 16




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

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

Ex. 1.3 (p.38) #1-12 odds in each with multiple parts
#3-16 odds in each with multiple parts


PC30.8
Demonstrate understanding of functions, relations, inverses
and their related equations resulting from reflections
through the: x-axis, y-axis, line y=x

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)= 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

Ex. 1.4 (p.51) #1-16 odds in questions with multiple parts
#4-20 odds in questions with multiple parts