Transformations of Graphs

Transformations are divided into translations, reflections, and stretches and shrinks. On this page, we will learn these three types.

1. Translations

A translation is a transformation which moves a graph

upward, downward, to the right, or to the left.

a. Vertical translations

There are two types of vertical translations: 1) move a graph upward, 2) shift a graph downward.

Let y = f (x) be an equation of a function, and k be a positive number.

i) Move the graph of y = f (x) upward

The graph of the function defined by y = f (x) + k will be obtained by moving the graph of the function y = f (x) upward by k units.

Example 1. Use the vertical translation to graph of the function defined by y = f (x) + 2.

The picture on the right shows the graph of a function defined by y = f (x). We don't need to know the exact equation.

In the equation, y = f (x) + 2, the positive number + 2 is located after f (x). This means that we shift the graph of y = f (x) upward by 2. Then we will get the red graph shown below.

Let y = f (x) be an equation of a function, and k be a positive number.

ii) Move the graph of y = f (x) downward

The graph of the function defined by y = f (x) - k will be obtained by moving the graph of the function y = f (x) downward by k units.

Example 2. Use the vertical translation to graph of the function defined by y = f (x) - 2. Let's use the same graph of y = f (x) in Example 1.

In the equation, y = f (x) - 2, the negative number - 2 is located after f (x). This means that we shift the graph of y = f (x) downward by 2. Then we will get the red graph shown below.

In a rectangular coordinate system, the values of y are located on the vertical axis (y-axis). On the y-axis, values increase in the upward direction, and decrease in the downward direction.

The value of k is the only difference between the equation of the original function y = f (x) and the new function y = f (x) + k. A positive value of k affects the value of y, causing the graph of y = f (x) to shift in the upward direction, resulting in the graph of y = f (x) + k.

Similarly, a negative value of -k affects the value of y, causing the graph of y = f (x) to shift in the downward direction, resulting in the graph of

y = f (x) - k.

Question 1. Graph the two functions using the graph of y = f (x) .