Transcript Transformations

1.5 Graphical Transformations

Represent translations algebraically and
graphically
Consider this…
How is the graph (x – 2)2 + (y+1)2 = 16
related to the graph of x2 + y2 = 16?
Some change is good!

Transformations - functions that map real numbers to real
numbers

Rigid transformations – leave the size and shape of a
graph unchanged, include horizontal translations, vertical
translations, reflections or any combination of these.

Non-rigid transformations – generally distort the shape of
a graph, include horizontal or vertical stretches and
shrinks.
Vertical and Horizontal Translations

Vertical translation – shift of the graph up or
down in the coordinate plane

Horizontal translation – shift of the graph left
or right in the coordinate plane
Exploration #1

Complete the activity on p. 132
–
–
With Partners
Use Whiteboards
Translations
Let c be a positive real number. Then the following transformations result
in translations of the graph of y = f(x)

Horizontal translations
y = f(x – c) a translation to the right by c units
y = f(x + c) a translation to the left by c units

Vertical translations
y = f(x) + c a translation up by c units
y = f(x) – c a translation down by c units
Describe how the graph of y = |x| can be transformed to
the graph of the given function:
a) y = |x – 4|
b) y = |x| + 2
Introducing Reflections

Reflections – the graphs of two functions are
symmetric with respect to some line

Complete Exploration #2 on p. 134
–
With Partners, Use Whiteboards
Reflections

Over the x-axis
–

Over the y-axis
–

Symbolically (x,y)  (x,-y)
Symbolically (x,y)  (-x,y)
Over the line y = x
–
Symbolically (x,y)  (y,x)
Reflections
Transformations that result in reflections of the graph y=f(x)

Across the x-axis:
y = - f(x)

Across the y-axis:
y = f(-x)
Express h(x) so that it represents the graph of
f(x) = x2 – 3 reflected over the x-axis? y-axis?
Find an equation for the reflection of
5 x  9 across each axis
f ( x) 
x2  3
Let’s think about this
What would happen if you reflected an even
function across the y-axis?
Vertical Stretches
When a function is multiplied by a number whose
absolute value is larger than 1, its graph is
stretched vertically
(Vertical Stretch by a factor of 3)
(Vertical Stretch by a factor of 2)
Vertical Shrinks
When a function is multiplied by a number between 0
and 1, its graph is shrunk vertically
(Vertical Shrink by a factor of 1/3)
(Vertical Shrink by a factor of 1/2)
Horizontal Shrinks
When x is multiplied by a number greater than 1, the
graph shrinks horizontally. This can look similar to
a vertical stretch, but it isn’t the same thing.
(Horizontal Shrink by a factor of ½ ) (Horizontal Shrink by a factor of ¼ )
Horizontal Stretches
When x is multiplied by a number between 0 and 1,
the graph stretches horizontally. This can look
similar to a vertical shrink, but it isn’t the same thing.
(Horizontal Stretch by a factor of 4)
(Horizontal Stretch by a factor of 3)
Finding equations for stretches and shrinks
Let y1 = f(x) = x3 – 16x. Find the equations for
the following transformations:
a)
A Vertical stretch by a factor of 3
a)
A Horizontal shrink by a factor of 1/2
Transform the given functions by (a) a vertical stretch
by a factor of 2 and (b) horizontal shrink by a factor of
1/3.
1) f(x) = |x + 2|
2) f(x) = x2 + x - 2
Combining Transformations
Use f(x) = x2 to perform each transformation:
a)
A horizontal shift 2 units to the right, a vertical stretch by a
factor of 3, and vertical translation 5 units up
a)
Apply the transformations in the reverse order
1.5 Homework


Part 1: page 139 – 140 Ex # 3-24 m. of 3
Part 2: page 140 Ex #39-42, 47-54 all