Graphing Tools: Vertical and Horizontal Scaling (Part 1)

(This page is Part 1. Click here for Part 2.)

Click here for a printable version of the discussion below.

You may want to review:

• Graphs of Functions
• Basic Models You Must Know
• Graphing Tools: Vertical and Horizontal Translations

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 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: $\,y=2{\text{e}}^{5x}\,$

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

• Multiply the previous $y$-values by $\,2\,,$ giving the new equation $\,y=2{\text{e}}^x\,.$ 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 $\,y=2{\text{e}}^{5x}\,.$ 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 $\,x\,$ 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: $$ \cssId{s30}{\bigl(x, \overset{\text{$y$-value}}{\overbrace{ f(x)}} \bigr)} $$ 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:

  $$ \cssId{s36}{\bigl(x, \overset{\text{$y$-value}}{\overbrace{ f(x)}} \bigr)} $$

• Compare the previous two ideas!

  To ‘graph the function $\,f\,$’ means to show all points of the form $\,\bigl(x,f(x)\bigr)\,.$

  To ‘graph the equation $\,y=f(x)\,$’ means to show all points of the form $\,\bigl(x,f(x)\bigr)\,.$

  These two requests mean exactly the same thing!

Ideas Regarding Vertical Scaling (Stretching/Shrinking)

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

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

  Points on the graph of $\,y=\frac13f(x)\,$ 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 $\,y=f(x)\,$ and asked about the graph of $\,y=3f(x)\,$:

interpretation of new equation:

Summary of Vertical Scaling

Let $\,k \gt 1\,.$

Start with the equation $\,y=f(x)\,.$ 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 $\,y=f(x)\,.$ 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 $\,(a,kb)\,$ on the graph of $\,y=kf(x)\,.$ This transformation type is formally called vertical scaling (stretching/shrinking).