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Translating Graphs Vertically

A translation is an operation that shifts a graph horizontally, vertically, or both.  The result is a graph that is the same shape and size, located in a different position.  The variables h and k are commonly used to represent the general form of a translation. Vertical translations are represented by the value for k, while h is the variable used to represent horizontal translations.

Notice that the k-value translates or shifts the graph on the y-axis. This is commonly referred to as a vertical translation.

Translating Graphs Horizontally

Notice that the h-value translates or shifts the graph onp the x-axis. This is commonly referred to as a horizontal translation.

Parent Functions

A family of functions is a group of functions with common characteristics.  The parent function is the simplest function with these characteristics. Begin with a parent function, make one or more translations or shifts, and you have a "family of functions."  By recognizing the shortcuts of a translation to the parent function, graphing becomes easy.

Let's look at the parent graph for linear functions.  That is, y = x and a related graph y = x + k.  Notice that the k-value translates or shifts the graph up the y-axis. This is commonly referred to as a vertical translation.

Writing Equations for Vertical Translations

You can write an equation for a vertical translation.  

Writing Equations for Horizontal Translations

Horizontal translations share some of the characteristics of vertical translations.  Remember we are using the variable h for horizontal translations.  We have already seen that when y = |x + h| the graph is translated h units to the left.  Then y = |x – h| would be translated h units to the right.

Diagonal Translations

Do you notice that the vertex of the absolute value function is located at the point (h, k)?  If so, you are now ready to graph and write equations of the parent function  y = |x|.