Transcript 2-6 Families of Functions

2-6 Families of
Functions
M11.D.2.1.2: Identify or graph functions, linear
equations, or linear inequalities on a coordinate
plane
Objectives
Translations
Stretches, Shrinks, and Reflections
Vocabulary
A family of functions is made up of functions with certain
common characteristics.
A parent function is the simplest function with these
characteristics. The equations of the functions in the family
resemble each other. So do the graphs.
A translation shifts the graph horizontally, vertically, or both.
It results in a graph of the same shape and size, but
possibly a different position.
Vertical Translation
Vertical Translation
a.
Describe the translation y = |x| – 2 and draw its
graph by translating the parent function.
y = |x| – 2 is a translation of y = |x| by
2 units downward. Each y-value for
y = |x| – 2 is 2 less than the corresponding
y-value for y = |x|.
b.
Write an equation for the translation of y = |x| up 8 units.
An equation that translates y = |x| up 8 units is
y = |x| + 8.
Horizontal Translation
Horizontal Translation
a. Describe the translation y = |x + 4| and draw its
graph by translating the parent function.
y = |x + 4| is a translation of y = |x| by
4 units left. Each x-value for
y = |x + 4| is 4 less than the corresponding
x-value for y = |x|.
b. Write an equation for the translation of y = |x| right
2 units.
An equation that translates y = |x| right 2 units is
y = |x – 2|.
Notes from the Book
Take a few minutes to write down the information inside the
orange box on page 95.
These notes may come in handy during your test.
Real World Example
Describe a possible translation of Figures M and N
in the design shown below.
Translate Figure M 2 units down, For Figure N, there are two
possible translations: 4 units down, or else 1 unit right and 2
units down.
Vocabulary
A vertical stretch multiplies all y-values by the same factor greater than 1,
thereby stretching the graph vertically.
A vertical shrink reduces the y-values by a factor between 0 and 1,
thereby compressing the graph vertically
More formally, for the parent function y = |x| and a number a,
a > 1, y = a|x| is a vertical stretch
0 < a < 1, y = a|x| is a vertical shrink
Graphing y = a|x|
Graphing y = a|x|
1
4
a. Describe and then draw the graph of y = |x|.
y = 1 |x| is a vertical shrink of y = |x| by
4
a factor of 1 . Each y-value for y = 1 |x| is
4
4
one-fourth the corresponding y-value
for y = |x|.
b. Write an equation for a vertical stretch of
y = |x| by a factor of 6.
A vertical stretch of y = |x| is y = 6|x|.
Vocabulary
A reflection in the x-axis changes y-values to their opposites.
When you change the y-values of the graph to their opposites, the graph
to their opposite, the graph reflects across the x-axis.
Graphing y = -a|x|
Write the equation for the graph.
The parent function is y = |x|.
The graph shows a vertical stretch of 3 units and
a reflection over the x-axis.
The equation of the graph is y = –3|x|.
Notes from the Book
Take a few minutes to write down the information inside the
orange box on page 97.
These notes may come in handy during your test.
Homework
Pg 97 & 98 # 1,5,8,12,15,17