Transcript Lesson 8.2

Lesson 8.2
Translating Graphs
•
To write equations to describe translations of the absolute-value
and squaring functions.
• To graph and recognize translations of the absolute value and
squaring functions.
• To explore the concept of a family of functions.
 In previous chapters you wrote many linear
and exponential functions. You used
points, y-intercepts, slope, starting values,
and constant multipliers to write equations
“from scratch.”
 In this chapter you will use transformations
to base functions such as y = |x| and y =
x2.
Translations of Functions
 First you’ll transform the absolute-value function by making
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changes to x.
Enter y = |x| into y1 and graph it on your calculator.
If you replace x with x-3 in the function
y = |x|, you get y = |x-3|. Enter y2=|x-3| and graph it.
How have you transformed the graph of
y = |x|?
Name the coordinates of the vertex of the graph of y = |x|.
Name the coordinates of the vertex of the graph of y =|x-3|.
How did these two points help verify the transformation you
just performed?
 Find a function for y2 that will translate the graph of left 4
units. What is the function? In the equation y = |x|, what
did you replace x with to get your new function?
 Write a function for y2 to create each graph below. Check
your work by graphing both y1 and y2.
 Next you will transform the absolute-value function by making
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changes to y.
Clear all of the functions. Enter y1= |x| and graph it.
If you replace y with y-3 in the function y= |x|, you get y3=|x|. Solve it for y and you get y=|x|+3. Enter y2=|x|+3.
Graph it.
Think of the graph of y= |x| as the original figure and the graph
of y=|x|+3 as its image. How have you transformed the graph of
y=|x|?
Name the coordinates of the vertex of both graphs. How do these
two points help verify the transformation you just found.
 Find a function y2 that will translate the graph of y=|x|
down 3 units. What is the function? In the function, y=|x|,
what did you replace y with to get your new function?
 Write a function for y2 to create each graph below. Check
your work by graphing both y1 and y2.
 Summarize what you have learned about translating the
absolute-value graph vertically and horizontally.
Definitions
 The most basic form of a function is called a parent function.
 By transforming the graph of a parent function, you can
create infinitely many new functions or a family of functions.
 y=|x-3| and y=|x| +3 are members of the absolute-value
family of functions.
Example A
 The graph of the parent
function y=x2 is shown
in bold.
 Its image after a
transformation is
shown in a thin line.
 Study the
transformation and
write the equation for
the transformed graph.
y  (4)  ( x  2)2
y  ( x  2)2  4
Example B
 The starting number of bacteria in a culture dish is unknown,
but the number grows by approximately 30% each hour.
After 4 hours, there are 94 bacteria present. Write an
equation to model this situation. Then find the starting
number of bacteria.
 The starting number is unknown, but you can find it by assuming that your
beginning with 94 bacteria, and then shifting back in time.
 If you began with 94 bacteria the function would be
y=94(1+0.30)x , where
 x represents the time elapsed in hours
 y represents the number of bacteria
The black graph
represents this function.
However, there were 94
bacteria after 4 hrs, not
at 0 hrs. So translate
the point (0,94) to
(4,94).
This changes the equation
to y=94(1+0.30)x-4