Graph Translations

Written by

Malcolm McKinsey

January 11, 2023

Fact-checked by

Paul Mazzola

Defining our terms

The abscissa is the x-coordinate, or the distance left or right from the y-axis that allows you to locate a point using a coordinate pair. Its partner is the ordinate, or y-coordinate. This is the distance above or below the x-axis.

Using the abscissa and ordinate, you can fix a point on the coordinate graph. Then, using translations, you can move the point. Translations are performed in three ways:

1. Shift - The graph of a function retains its size and shape but moves (slides) to a new location on the coordinate grid

2. Scale - The size and shape of the graph of a function is changed

3. Reflection - A mirror image of the graph of a function is generated across either the x-axis or y-axis

Shifting, scaling and reflecting are three methods of producing translations for basic graphing functions you have already learned. Knowing how to shift, scale or reflect these graphs makes you a stronger mathematics student and produces many variations on the original graphs of common functions.

Reviewing common functions

In order to translate any of the common graphed functions, you need to recall and be fluent with the seven common functions themselves, presented here alphabetically because they are all equally important:

1. Absolute Value Function: $y=\left|x\right|$

2. Constant Function: $y=c$

3. Cubic Function: $y={x}^{3}$

4. Greatest Integer Function or Floor Function: $y=\left[x\right]$

5. Linear Function or Identity Function: $y=x$

6. Quadratic Function or Squaring Function: $y={x}^{2}$

7. Square Root Function: $y=\sqrt{x}$

By concentrating on the original seven functions and the way they appear when graphed, you will soon develop an awareness of how each of the three translations affects the original graphed function.

Rather than burden your brain by trying to memorize the configurations of some 28 different graphs (each original and three translations), concentrate on the changes each translation provides to its original function.

The mathematics will not seem so intimidating. Rather than get lost in the details, you will come to know the graphs of all seven functions, and their translations, as a group. You will be able to do better mathematics faster, since you will save time not having to plot out individual points. Instead you will learn to recognize a given graph as, for example, the reflection of a graph of a cubic function.

Here are graphs of the seven functions. Make sure you are familiar with the shape and direction of each graph.

Translations of the common functions

Each of the seven graphed functions can be translated by shifting, scaling, or reflecting:

1. Shift - A rigid translation, the shift does not change the size or shape of the graph of the function. A shift will move the graph to a new location on the coordinate system. To move vertically, a constant is added or subtracted from each y-coordinate. The x-coordinate is unchanged. To move horizontally, a constant is added or subtracted to each x-coordinate. The y-coordinate is unchanged. Both vertical and horizontal shifts can be shown as a single expression.

2. Scale - A non-rigid translation, the scale changes the size and shape of the graph of a function. Scaling can multiply or divide the coordinates (x, y), which changes the appearance and location of the graph. Vertical scaling affects only the y-coordinates; horizontal scaling affects the x-coordinates. As with the shift, scale will not affect the other coordinate, and both vertical and horizontal scaling can be shown as a single expression.

3. Reflection - A rigid translation, the reflection is achieved by multiplying one coordinate by -1. To reflect across the y-axis, the x-coordinate is multiplied to get -x. To reflect or flip across the x-axis, multiply everything by -1. Yes, the entire function is multiplied by -1: $f(x)\times -1=-f(x)$. The graph is flipped "upside down."

A single equation

All three functions can be combined into a single equation. Before you pick up this powerful tool, though, make certain you know all its parts:

• a = the vertical stretch or compression (scale), for which $\left|a\right|>1$ stretches the graph, $\left|a\right|<1$ compresses the graph, and $a<0$ reflects or flips the graph upside down

• b = the horizontal stretch or compression (scale), for which $\left|b\right|>1$ compresses the graph, $\left|b\right|<1$ stretches the graph, and $b<0$ reflects or flips the graph left and right

• c = the horizontal shift, for which $c<0$ shifts everything right and $c>0$ shifts everything left

• d = the vertical shift, for which $d>0$ moves everything up and $d<0$ moves everything down

And now, the powerful, single equation:

If you do not have a value for one of the variables, your equation still works. Do not replace the missing variable with a 0. For example, for the cubic function $y={x}^{3}$, the original graph looks like this:

[insert graph of y=x^3]

The translation equation, $y=(-1({x}^{3}+1))=1$, which has -1 for b, 1 for c, and 1 for d but has no value for a, looks like this:

[insert graph showing both the original function and translated equation]

Incorrectly putting a 0 in for the a value for a produces this equation: $y=0(-1({x}^{3}+1))+1$ and looks like this: [insert graph showing both original function and straight line from equation above]

Clearly this is an entirely different function, unrelated to your original cubic function. When in doubt, leave it out.

Examples of translation functions

$y=f(x)$ produces no translation; no values for a, b, c or d are shown.

$y=f(x+2)$ produces a horizontal shift to the left, because the +2 is the c value from our single equation. It is added to the x-value. For horizontal shifts, positive c values shift the graph left and negative c values shift the graph right.

$y=f(x)+2$ produces a vertical translation, because the +2 is the d value. It shifts the entire graph up for positive values of d and down for negative values of d.

$y=f(x-2)+3$ gives us values for both c and d, so the translation moves 2 units right (negative c) and three units up (positive d).

$y=3\times f(x)$ produces a change of scale, because the absolute value of 3 (the a value in our single equation) stretches the graph.

$y=-1\times f(x)=-f(x)$ indicates we have multiplied everything by -1; this produces a reflection across the x-axis, without changing x-axis values.

$y=f(2x)$ shows our x-value multiplied, which means we have scaled the original function horizontally, which shrinks the graph. Horizontal changes are inverse. Multiplying by 2 actually divides every x-value by 2 to produce the y-value.

$y=f(-1\times x)=f(-x)$ shows we multiplied only the x-value times -1; this will reflect the graph across the y-axis, without changing y-values.

Cautionary tales

Besides the issue of putting in 0s in the single equation when unnecessary, other stumbling blocks to correctly graphing translations of functions are common. Perhaps most common is keeping straight how some values are inverse, some seem backwards, and some are exactly as they appear.

To move a graph up, we add a positive value to the y-value. To move a graph down, we add a negative value to the y-value.

To move a graph right, we add a negative value to the x-value. To move a graph left, we add a positive value to the x-value.

To stretch a graph in the y-axis, we multiply the whole function times any number n such that $n>1$. This will cause, for example, a vertical parabola to "open up." To compress a graph, we multiply the whole function times any number n such that $0<n<1$. This will cause the same vertical parabola to "close up."

To stretch a graph in the x-axis, we multiply only the x-values times a constant. Larger values compress the graph and smaller values ($0<n<1$) stretch the graph. This is inverse of stretching the graph in the y-axis.

To reflect a graph across the x-axis, everything is multiplied by -1.

To reflect a graph across the y-axis, only the x-value is multiplied by -1.