Graphing Tools: Vertical and Horizontal Translations

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You may want to review:

• Graphs of Functions
• Basic Models You Must Know

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.

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:

  $$ \cssId{s35}{\bigl(x, \overset{\text{y-value}}{\overbrace{ f(x)}} \bigr)} $$

  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: $$ \cssId{s41}{\bigl(x, \overset{\text{y-value}}{\overbrace{ f(x)}} \bigr)} $$

• 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 Translations (Moving Up/Down)

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

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

  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 $\,y=f(x)+3\,$:

interpretation of new equation:

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 $\,\bigl(x,f(x)\bigr)\,.$

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

• How can we locate these desired points $\,\bigl(x,f(x+3)\bigr)\,$?

  

  First, go to the point $\,\color{red}{\bigl(x+3\,,\,f(x+3)\bigr)}\,$ 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.

  Move this point $\,\color{purple}{3}\,$ units to the left. Thus, the $y$-value stays the same, but the $x$-value is decreased by $\,3\,.$ This gives the desired point $\,\color{green}{\bigl(x,f(x+3)\bigr)}\,.$

  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 $\,y=f(x+3)\,$:

interpretation of new equation:

• 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.

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 $\,y\,$ (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 $\ldots$

Make Sure You See The Difference!

Vertical Translations:
Going from $\,y=f(x)\,$ to $\,y = f(x) \pm c\,$

Horizontal Translations:
Going from $\,y = f(x)\,$ to $\,y = f(x\pm c)\,$

Make sure you see the difference between (say) $\,y = f(x) + 3\,$ and $\,y = f(x+3)\,$!

In the case of $\,y = f(x) + 3\,,$ the $\,3\,$ is ‘on the outside’; we're dropping $\,x\,$ in the $\,f\,$ box, getting the corresponding output, and then adding $\,3\,$ to it. This is a vertical translation.

In the case of $\,y = f(x + 3)\,,$ the $\,3\,$ is ‘on the inside’; we're adding $\,3\,$ to $\,x\,$ before dropping it into the $\,f\,$ box. This is a horizontal translation.

Examples