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GRAPHING TOOLS:
REFLECTIONS and the ABSOLUTE VALUE TRANSFORMATION
Jump right to the exercises!
Click here for a printable (pdf) version of the discussion below.
There are things that you can DO to an equation of the form
y=f(x
)
that will change the graph in a variety of ways.
For example, you can move the graph up or down, left or right,
reflect about the x or
y axes, stretch or shrink vertically or horizontally.
An understanding of these transformations makes it easy to graph a wide variety of functions,
by starting with a "basic model" and then applying a
sequence of transformations to change it to the desired function.
In this discussion, we will explore reflecting about the x-axis and the
y-axis, and the absolute value transformation.
When you finish studying this lesson, you should be able to do a problem like this:
GRAPH: y=-|ln
(-x)|
- Start with the graph of y=ln(x
) . (This is the "basic model".)
- Replace every x by -x , giving the new equation
y=ln(-
x) .
This reflects the graph about the y-axis.
- Take the absolute value of the previous y-values, giving the new equation
y=|ln(
-x)| .
This takes any part of the graph below the x-axis, and reflects it about the x-axis.
Any part of the graph on or above the x-axis remains the same.
- Multiply the previous y-values by -1 , giving the new equation
y=-|ln
(-x)| .
This reflects the graph about the x-axis.
Here are ideas that are needed to understand graphical transformations.
FIRST, SOME IDEAS REGARDING FUNCTIONS AND THE GRAPH OF A FUNCTION:
- A function is a rule:
it takes an input, and gives a unique output.
- If x is the input to a function f ,
then the unique output is called f(x) (which is read as " f of x ").
- The graph of a function is a picture of all of its (input,output) pairs.
We put the inputs along the horizontal axis (the x-axis),
and the outputs along the vertical axis (the y-axis).
- Thus, the graph of a function f is a picture of all points of the form
( x,
f(x)
⏞
y-value
)
.
Here, x is the input, and f(x) is the corresponding output.
- The equation y=f(x
) is an equation in two variables, x and y .
A solution is a choice for x and a choice for y that makes the equation true.
Of course, in order for this equation to be true, y must equal f(x) .
Thus, solutions to the equation y=f(x
) are points of the form
( x,
f(x)
⏞y-value
) .
- Compare the previous two ideas!
To "graph the function f " means to show all points of the form
(x,f(
x)) .
To "graph the equation y=f(x
) " means to show all points of the form
(x,f(
x)) .
These two requests mean exactly the same thing!
IDEAS REGARDING REFLECTING ABOUT THE X-AXIS:
- Points on the graph of y=f(x
) are of the form (x,f(
x)) .
Points on the graph of y=-f(x
) are of the form
(x,-f(
x)) .
Thus, the graph of y=-f(x
) is found by taking the graph of y=f(x
) and multiplying the y-values by -1 .
This reflects the graph about the x-axis.
- Transformations involving y work the way you would expect them to work—they are intuitive.
- Here is the thought process you should use when you are given the graph of
y=f(x
)
and asked about the graph of
y=-f(x
) :
original equation: y=f(x
)
new equation: y=-f(x
)
interpretation of new equation:
y
⏟
the new y-values
=
⏟
are
-
⏟
-1 times
f(x)
⏟
the previous y-values
-
In reflection about the x-axis, a point
(a,b)
on the graph of
y=f(x
)
moves to a point
(a,-b)
on the graph of
y=-f(x
) .
IDEAS REGARDING REFLECTING ABOUT THE Y-AXIS:
- Points on the graph of y=f(x
) are of the form (x,f(
x)) .
Points on the graph of y=f(-x
) are of the form
(x,f(-
x)) .
-
How can we locate these desired points (x,f(
-x)) ?
First, go to the point
(-x , f(-x)) on the
graph of y=f(x
) .
This point has the y-value that we want, but it has the wrong x-value.
The x-value of this point is -x , but the desired x-value is just x .
Thus, the current x-value must be multiplied by -1 ; that is, each x-value must be
sent to its opposite.
The y-value remains the same.
This causes the point to reflect about the y-axis. This gives the desired point
(x,f(
-x)) .
Thus, the graph of y=f(-x
) is the same as the graph of
y=f(x
) , except that it has been reflected about the y-axis.
- Here is the thought process you should use when you are given the graph of
y=f(x
)
and asked about the graph of
y=f(-x
) :
original equation: y=f(x
)
new equation: y=f(-x
)
interpretation of new equation:
y=
f(
-x
⏟
replace x by -x
)
Replacing every x by
-x in an equation causes the graph to
be reflected about the y-axis.
- In reflection about the y-axis,
a point (a,b)
on the graph of y=f(x
) moves to a point (-a,b) on the graph of
y=f(-x
) .
IDEAS REGARDING THE ABSOLUTE VALUE TRANSFORMATION:
- Points on the graph of y=f(x
) are of the form (x,f(
x)) .
Points on the graph of y=|f(
x)| are of the form
(x,| f(
x)|) .
- Thus, the graph of y=|f(x
)| is found by taking the graph of y=f(x
) and taking the absolute value of the y-values.
Points with positive y-values stay the same, since the absolute value of a positive number is itself.
That is, point above the x-axis don't change.
Points with y = 0 stay the same, since the absolute value of zero is itself.
That is, point on the x-axis don't change.
Points with negative y-values will change, since taking the absolute value of a negative number makes it positive.
That is, any point below the x-axis reflects about the x-axis.
These actions are summarized by saying that "any part of the graph below the x-axis flips up".
- Here is the thought process you should use when you are given the graph of
y=f(x
)
and asked about the graph of
y=|f(x
)| :
original equation: y=f(x
)
new equation: y=|f(x)|
interpretation of new equation:
y
⏟
the new y-values
=
⏟
are
| f(x)|
⏟
the absolute value of the previous y-values
-
In the absolute value transformation, a point
(a,b)
on the graph of
y=f(x
)
moves to a point
(a,|b|) on the graph of
y=|f(x
)| .
reflecting about the x-axis:
going from
y
=
f
(
x
)
to
y=
−
f(x)
reflecting about the y-axis:
going from
y
=
f
(
x
)
to
y=
f(−x)
absolute value transformation:
going from
y
=
f
(
x
)
to
y=
|
f(x)
|
Any part of the graph on or above the x-axis stays the same;
any part of the graph below the x-axis flips up!
EXAMPLES:
Start with
y=x .
Reflect about the x-axis.
What is the new equation?
Answer:
y=−x
Start with
y=e
x .
Reflect about the y-axis.
What is the new equation?
Answer:
y=e
−x
Suppose
(
a
,
b
)
is a point on the graph of
y=x3 .
Then, what point is on the graph of
y=
|x3|
?
Answer:
(
a
,
|
b
|
)
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.