There are things that you can DO to an equation of the form
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$\,y=f(x)\,$
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
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$\,x\,$ 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 stretching and shrinking a graph, both vertically and horizontally.
When you finish studying this lesson, you should be able to do a problem like this:
GRAPH:
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$\,y=2{\text{e}}^{5x}\,$

Start with the graph of
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$\,y={\text{e}}^x\,$.
(This is the ‘basic model’.)

Multiply the previous $\,y$values by $\,2\,$,
giving the new equation
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$\,y=2{\text{e}}^x\,$.
This produces a vertical stretch, where the $\,y$values on the graph get multiplied by $\,2\,$.
 Replace every $\,x\,$ by
$\,5x\,$, giving the new equation
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$\,y=2{\text{e}}^{5x}\,$.
This produces a horizontal shrink, where the $\,x$values on the graph get divided by $\,5\,$.
Here are ideas that are needed to understand graphical transformations.
IDEAS REGARDING FUNCTIONS AND THE GRAPH OF A FUNCTION

A function is a rule:
it takes an input, and gives a unique output.

If
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$\,x\,$ 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
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$\,\bigl(x,
\overset{\text{yvalue}}{\overbrace{
f(x)}}
\bigr) \,$.
Here, $\,x\,$ is the input, and $\,f(x)\,$ is the corresponding output.

The equation
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$\,y=f(x)\,$ 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
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$\,\bigl(x,
\overset{\text{yvalue}}{\overbrace{
f(x)}}
\bigr) \,$.

Compare the previous two ideas!
To ‘graph the function $\,f\,$’ means to show all points of the form
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$\,\bigl(x,f(x)\bigr)\,$.
To ‘graph the equation $\,y=f(x)\,$’ means to show all points of the form
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$\,\bigl(x,f(x)\bigr)\,$.
These two requests mean exactly the same thing!
IDEAS REGARDING VERTICAL SCALING (STRETCHING/SHRINKING)
 Points on the graph of
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$\,y=f(x)\,$ are of the form $\,\bigl(x,f(x)\bigr)\,$.
Points on the graph of
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$\,y=3f(x)\,$ are of the form $\,\bigl(x,3f(x)\bigr)\,$.
Thus, the graph of $\,y=3f(x)\,$ is found by taking the graph of $\,y=f(x)\,$,
and multiplying the $\,y$values by $\,3\,$.
This moves the points farther from the $\,x$axis, which makes the graph steeper.
 Points on the graph of
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$\,y=f(x)\,$ are of the form $\,\bigl(x,f(x)\bigr)\,$.
Points on the graph of
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$\,y=\frac13f(x)\,$ are of the form $\,\bigl(x,\frac13f(x)\bigr)\,$.
Thus, the graph of $\,y=\frac13f(x)\,$ is found by taking the graph of $\,y=f(x)\,$,
and multiplying the $\,y$values by $\,\frac13\,$.
This moves the points closer to the $\,x$axis, which makes the graph flatter.

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
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$\,y=f(x)\,$
and asked about the graph of
$\,y=3f(x)\,$:
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$$
\begin{align}
\text{original equation:} &\quad y=f(x)\cr\cr
\text{new equation:} &\quad y=3f(x)
\end{align}
$$
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$$
\begin{gather}
\text{interpretation of new equation:}\cr\cr
\overset{\text{the new yvalues}}{\overbrace{
\strut\ \ y\ \ }}
\overset{\text{are}}{\overbrace{
\strut\ \ =\ \ }}
\overset{\quad\text{three times}\quad}{\overbrace{
\strut \ \ 3\ \ }}
\overset{\qquad\text{the previous yvalues}\quad}{\overbrace{
\strut\ \ f(x)\ \ }}
\end{gather}
$$

Summary of vertical scaling:
Let $\,k \gt 1\,$.
Start with the equation
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$\,y=f(x)\,$.
Multiply the previous $\,y\,$values by $\,k\,$, giving the new equation
$\,y=kf(x)\,$.
The $\,y$values are being multiplied by a number greater than $\,1\,$, so they move farther from the $\,x$axis.
This makes the graph steeper, and is called a vertical stretch.
Let $\,0 \lt k \lt 1\,$.
Start with the equation
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$\,y=f(x)\,$.
Multiply the previous $\,y\,$values by $\,k\,$, giving the new equation
$\,y=kf(x)\,$.
The $\,y$values are being multiplied by a number between $\,0\,$ and $\,1\,$, so they move closer to the $\,x$axis.
This makes the graph flatter, and is called a vertical shrink.
In both cases, a point $\,(a,b)\,$ on the graph of $\,y=f(x)\,$ moves to a point $\,(a,k\,b)\,$
on the graph of $\,y=kf(x)\,$.
This transformation type is formally called vertical scaling (stretching/shrinking).
IDEAS REGARDING HORIZONTAL SCALING (STRETCHING/SHRINKING)

Points on the graph of
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$\,y=f(x)\,$ are of the form $\,\bigl(x,f(x)\bigr)\,$.
Points on the graph of $\,y=f(3x)\,$ are of the form $\,\bigl(x,f(3x)\bigr)\,$.

How can we locate these desired points $\,\bigl(x,f(3x)\bigr)\,$?
First, go to the point
$\,\color{red}{\bigl(3x\,,\,f(3x)\bigr)}\,$
on the graph of
$\,\color{red}{y=f(x)}\,$.
This point has the $\,y$value that we want, but it has the wrong $\,x$value.
The $\,x$value of this point is $\,3x\,$, but the desired $\,x$value is just $\,x\,$.
Thus, the current
$\,\color{purple}{x}$value must be divided by $\,\color{purple}{3}\,$; the $\,\color{purple}{y}$value remains the same.
This gives the desired point
$\,\color{green}{\bigl(x,f(3x)\bigr)}\,$.
Thus, the graph of $\,y=f(3x)\,$ is the same as the graph of $\,y=f(x)\,$,
except that the $\,x$values have been divided by $\,3\,$ (not multiplied by $\,3\,$, which you might expect).
Notice that dividing the $\,x$values by $\,3\,$ moves them closer to the $\,y$axis; this is called a horizontal shrink.

Transformations involving $\,x\,$
do NOT work the way you would expect them to work!
They are counterintuitive—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 $\,y=f(3x)\,$:
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$$
\begin{align}
\text{original equation:} &\quad y=f(x)\cr\cr
\text{new equation:} &\quad y=f(3x)
\end{align}
$$
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$$
\begin{gather}
\text{interpretation of new equation:}\cr\cr
y = f(
\overset{\text{replace x by 3x}}{\overbrace{
\ \ 3x\ \ }}
)
\end{gather}
$$
Replacing every $\,x\,$ by
$\,3x\,$ in an equation
causes the $\,x$values in the graph to be DIVIDED by $\,3\,$.

Summary of horizontal scaling:
Let $\,k\gt 1\,$.
Start with the equation
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$\,y=f(x)\,$.
Replace every $\,x\,$ by $\,k\,x\,$ to
give the new equation $\,y=f(k\,x)\,$.
This causes the $\,x$values on the graph to be DIVIDED by $\,k\,$, which moves the points closer to the $\,y$axis.
This is called a horizontal shrink.
A point $\,(a,b)\,$ on the graph of $\,y=f(x)\,$ moves to a point $\,(\frac{a}{k},b)\,$ on the graph of
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$\,y=f(k\,x)\,$.
Additionally:
Let $\,k\gt 1\,$.
Start with the equation
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$\,y=f(x)\,$.
Replace every $\,x\,$ by $\,\frac{x}{k}\,$ to
give the new equation $\,y=f(\frac{x}{k})\,$.
This causes the $\,x$values on the graph to be MULTIPLIED by $\,k\,$, which moves the points farther away from the $\,y$axis.
This is called a horizontal stretch.
A point $\,(a,b)\,$ on the graph of $\,y=f(x)\,$ moves to a point $\,(k\,a,b)\,$ on the graph of
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$\,y=f(\frac{x}{k})\,$.
This transformation type is formally called horizontal scaling (stretching/shrinking).

DIFFERENT WORDS USED TO TALK ABOUT TRANSFORMATIONS INVOLVING $\,y\,$ and $\,x\,$
Notice that different words are used when talking about transformations involving
$\,y\,$, and transformations involving $\,x\,$.
For transformations involving
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$\,y\,$
(that is, transformations that change the $\,y$values of the points),
we say:
DO THIS to the previous $\,y$value.
For transformations involving
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$\,x\,$
(that is, transformations that change the $\,x$values of the points),
we say:
REPLACE the previous $\,x$values by $\ldots$
MAKE SURE YOU SEE THE DIFFERENCE!
vertical scaling:
going from
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$\,y=f(x)\,$
to
$\,y = kf(x)\,$ for $\,k\gt 0$
horizontal scaling:
going from
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$\,y = f(x)\,$
to
$\,y = f(k\,x)\,$ for $\,k\gt 0$
Make sure you see the difference between (say)
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$\,y = 3f(x)\,$
and
$\,y = f(3x)\,$!
In the case of
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$\,y = 3f(x)\,$, the $\,3\,$ is ‘on the outside’;
we're dropping $\,x\,$ in the $\,f\,$ box, getting the corresponding output, and then multiplying by $\,3\,$.
This is a vertical stretch.
In the case of
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$\,y = f(3x)\,$, the $\,3\,$ is ‘on the inside’;
we're multiplying $\,x\,$ by $\,3\,$ before dropping it into the $\,f\,$ box.
This is a horizontal shrink.
EXAMPLES:
Question:
Start with $\,y = f(x)\,$.
Do a vertical stretch; the $\,y$values on the graph should be multiplied by $\,2\,$.
What is the new equation?
Solution:
This is a transformation involving $\,y\,$; it is intuitive.
You must multiply the previous $\,y$values by $\,2\,$.
The new equation is:
$\,y = 2f(x)\,$
Question:
Start with $\,y = f(x)\,$.
Do a horizontal stretch; the $\,x$values on the graph should get multiplied by $\,2\,$.
What is the new equation?
Solution:
This is a transformation involving $\,x\,$; it is counterintuitive.
You must replace every $\,x\,$ in the equation by $\,\frac{x}{2}\,$.
The new equation is:
$\,y = f(\frac{x}{2})\,$
Question:
Start with $\,y = x^3\,$.
Do a vertical shrink, where $\,(a,b) \mapsto (a,\frac{b}{4})\,$.
What is the new equation?
Solution:
This is a transformation involving $\,y\,$; it is intuitive.
You must multiply the previous $\,y$values by $\frac 14\,$.
The new equation is:
$\,y = \frac14 x^3\,$
Question:
Suppose $\,(a,b)\,$ is a point on the graph of $\,y = f(x)\,$.
Then, what point is on the graph of $\,y = f(\frac{x}{3})\,$?
Solution:
This is a transformation involving $\,x\,$; it is counterintuitive.
Replacing every $\,x\,$ by $\,\frac{x}{3}\,$ in the equation causes the $\,x$values on the graph to be multiplied by $\,3\,$.
Thus, the new point is $\,(3a,b)\,$.