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GRAPHING TOOLS:
VERTICAL AND HORIZONTAL TRANSLATIONS
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 moving a graph up/down (vertical translations)
and moving a graph left/right (horizontal translations).
When you finish studying this lesson, you should be able to do a problem like this:
GRAPH:
y=
(x-3)
2
+5
- Start with the graph of y=x
2 . (This is the "basic model".)
- Add 5 to the previous y-values, giving the new equation
y=x
2+5 .
This moves the graph UP 5 units.
- Replace every x by
x-3 , giving the new equation
y=(x
-3)2
+5 .
This moves the graph to the RIGHT 3 units.
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 VERTICAL TRANSLATIONS (MOVING UP/DOWN):
- Points on the graph of y=f(x
) are of the form (x,f(
x)) .
Points on the graph of y=f(x
)+3 are of the form
(x,f(
x)+3) .
Thus, the graph of y=f(x
)+3 is the same as the graph of
y=f(x
) , shifted UP three units.
- 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
)+3 :
original equation: y=f(x
)
new equation: y=f(x
)+3
interpretation of new equation:
y
⏟
the new y-values
=
⏟
are
f(x)
⏟
the previous y-values
+
3
⏟
with 3 added to them!
-
Summary of vertical translations:
Let p be a positive number.
Start with the equation
y=f(x
) .
Adding p to the previous y-values gives the new equation
y=f(x
)+p .
This shifts the graph UP p units.
A point (a,b)
on the graph of y=f(x
) moves to a point (a,b+
p) on the graph of
y=f(x
)+p .
Additionally:
Start with the equation
y=f(x
) .
Subtracting p from the previous y-values gives the new equation
y=f(x
)-p .
This shifts the graph DOWN p units.
A point (a,b)
on the graph of y=f(x
) moves to a point (a,b-
p) on the graph of
y=f(x
)-p .
This transformation type (shifting up and down) is formally called vertical translation.
IDEAS REGARDING HORIZONTAL TRANSLATIONS (MOVING LEFT/RIGHT):
- Points on the graph of y=f(x
) are of the form (x,f(
x)) .
Points on the graph of y=f(x
+3) are of the form
(x,f(
x+3)) .
-
How can we locate these desired points (x,f(
x+3)) ?
First, go to the point
(x+3 , f(
x+3)) on the
graph of y=f(x
) .
This point has the y-value that we want, but it has the wrong x-value.
Move this point 3 units to the left.
Thus, the y-value stays the same, but the x-value is decreased by 3 .
This gives the desired point
(x,f(
x+3)) .
Thus, the graph of y=f(x
+3) is the same as the graph of
y=f(x
) , shifted LEFT three units.
Thus, replacing x by
x+3 moved the graph LEFT
(not right, as might have been expected!)
- Transformations involving x do NOT work the way you would expect them to work—they are counter-intuitive—they are against your intuition.
- 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+3
) :
original equation: y=f(x
)
new equation: y=f(x
+3)
interpretation of new equation:
y=
f(
x+3
⏟
replace x by x+3
)
Replacing every x by
x+3 in an equation moves the
graph 3 units TO THE LEFT.
-
Summary of horizontal translations:
Let p be a positive number.
Start with the equation
y=f(x
) .
Replace every x by
x+p to
give the new equation
y=f(x
+p) .
This shifts the graph LEFT p units.
A point (a,b)
on the graph of y=f(x
) moves to a point (a-p,b) on the graph of
y=f(x
+p) .
Additionally:
Start with the equation
y=f(x
) .
Replace every x by
x-p to
give the new equation
y=f(x
-p) .
This shifts the graph RIGHT p units.
A point (a,b)
on the graph of y=f(x
) moves to a point (a+p,b) on the graph of
y=f(x
-p) .
This transformation type (shifting left and right) is formally called horizontal translation.
Notice that different words are used when talking about transformations involving
y , and
transformations involving x .
For transformations involving y (that is, transformations that change the y-values of the points),
we say:
DO THIS to the previous y-value.
For transformations involving x (that is, transformations that change the x-values of the points),
we say:
REPLACE the previous x -values by ... ..
vertical translations:
going from
y
=
f
(
x
)
to
y
=
f
(
x
)
±
c
horizontal translations:
going from
y
=
f
(
x
)
to
y
=
f
(
x
±
c
)
EXAMPLES:
Start with
y
=
f
(
x
)
.
Move the graph TO THE RIGHT 2 .
What is the new equation?
Solution:
This is a transformation involving x ; it is counter-intuitive.
You must replace every x by
x-2 .
The new equation is:
y
=
f
(
x
-
2
)
Start with
y
=
x
2
.
Move the graph DOWN 3 .
What is the new equation?
Solution:
This is a transformation involving y ; it is intuitive.
You must subtract 3 from the previous y-value.
The new equation is:
y
=
x
2
-
3
Let
(
a
,
b
)
be a point on the graph of
y
=
f
(
x
)
.
Then, what point is on the graph of
y
=
f
(
x
+
5
)
?
Answer:
(
a
-
5
,
b
)
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.