The Algebra Teacher\'s Guide to Reteaching Essential Concepts and Skills

Teaching Notes 8.6: Describing Vertical Shifts of the Graph of

a Function

Using the equation that describes the graph of a function offers important information about

vertical shifts. Mistakes often occur when students misinterpret the numbers given in the

equation.

1. Show the graphs of the following basic functions to your students:

Identity:f(x)=x Squaring:f(x)=x^2 Cubing:f(x)=x^3

Square root:f(x)=

√

x Absolute value:f(x)=|x| Reciprocal:f(x)=

1

x

2. Explain that adding a positive number to a function moves the graph of a function up.
   Adding a negative number to a function moves the graph of a function down. For example,
   the graph off(x)=|x|+3 is the same size and shape as the graph off(x)=|x|but the graph
   is moved up 3 units. Similarly, the graph off(x)=x^2 −2 is the same size and shape as the
   graph off(x)=x^2 , except the graph is moved down 2 units. To illustrate these concepts, you
   may wish to sketch the graph of each function by plotting points.


3. Review the information and example on the worksheet with your students. Note that stu-
   dents must first identify the basic function and then find the value ofd. Also note that the
   domains of the functions are restricted so that the functions are defined.

EXTRA HELP:

Vertical shifts do not change the shape of the graph of a function.

ANSWER KEY:

The graph shows the following:

(1)Reciprocal function shifted up 3 units. (2)Cubing function shifted down 2 units.

(3)Squaring function shifted down 4 units. (4)Absolute value function was not shifted.

(5)Square root function shifted up 2 units. (6)Identity function shifted down 1 unit.

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(Challenge)The domain of the function does not change. The range may change, depending

on the function. For example, the range off(x)=x^3 isthesameastherangeoff(x)=x^3 +1.

But the range ofg(x)=x^2 is different from the range ofg(x)=x^2 +1.

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284 THE ALGEBRA TEACHER’S GUIDE