GRAPHING TOOLS:
VERTICAL AND HORIZONTAL SCALING

There are things that you can DO to an equation of the form [beautiful math coming... please be patient] $\,y=f(x)\,$
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 [beautiful math coming... please be patient] $\,x\,$ 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: [beautiful math coming... please be patient] $\,y=2{\text{e}}^{5x}\,$

Here are ideas that are needed to understand graphical transformations.

IDEAS REGARDING FUNCTIONS AND THE GRAPH OF A FUNCTION

IDEAS REGARDING VERTICAL SCALING (STRETCHING/SHRINKING)
IDEAS REGARDING HORIZONTAL SCALING (STRETCHING/SHRINKING)
  • Points on the graph of [beautiful math coming... please be patient] $\,y=f(x)\,$ 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 $\,\bigl(3x\,,\,f(3x)\bigr)\,$ 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 $\,3x\,$, but the desired $\,x$-value is just $\,x\,$.
    Thus, the current $\,x$-value must be divided by $\,3\,$; the $\,y$-value remains the same.
    This gives the desired point $\,\bigl(x,f(3x)\bigr)\,$.

    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 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 $\,y=f(3x)\,$: [beautiful math coming... please be patient] $$ \begin{align} \text{original equation:} &\quad y=f(x)\cr\cr \text{new equation:} &\quad y=f(3x) \end{align} $$ [beautiful math coming... please be patient] $$ \begin{gather} \text{interpretation of new equation:}\cr\cr y = f( \overset{\text{replace x by 3x}}{\overbrace{ \ \ 3x\ \ }} ) \end{gather} $$ 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 [beautiful math coming... please be patient] $\,y=f(x)\,$.
    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 [beautiful math coming... please be patient] $\,y=f(k\,x)\,$.

    Additionally:
    Let $\,k\gt 1\,$.
    Start with the equation [beautiful math coming... please be patient] $\,y=f(x)\,$.
    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 [beautiful math coming... please be patient] $\,y=f(\frac{x}{k})\,$.

    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 [beautiful math coming... please be patient] $\,y\,$
(that is, transformations that change the $\,y$-values of the points), we say:

DO THIS to the previous $\,y$-value.

For transformations involving [beautiful math coming... please be patient] $\,x\,$
(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   [beautiful math coming... please be patient] $\,y=f(x)\,$   to   $\,y = kf(x)\,$   for   $\,k\gt 0$

horizontal scaling:
going from   [beautiful math coming... please be patient] $\,y = f(x)\,$   to   $\,y = f(k\,x)\,$   for   $\,k\gt 0$

Make sure you see the difference between (say) [beautiful math coming... please be patient] $\,y = 3f(x)\,$ and $\,y = f(3x)\,$!

In the case of [beautiful math coming... please be patient] $\,y = 3f(x)\,$, 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 [beautiful math coming... please be patient] $\,y = f(3x)\,$, 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 counter-intuitive.
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 counter-intuitive.
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 $\,(3a,b)\,$.
Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Reflections and the Absolute Value Transformation


On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
(MAX is 64; there are 64 different problem types.)