Pre-Calculus 11 Absolute Value Functions

Lesson Focus: To create a table of values for y = |f(x)|, given a table of values for y = f(x); to sketch the graph

of y = |f(x)| and determining its intercept(s), domain, and range; to generalize a rule for writing absolute value

functions in piecewise notation.

an absolute value function a function that involves the absolute value of a variable

i.e. 23 xy

consider the functions xxgxxf and

complete the table below for each function and then graph each function on the axes provided

consider the functions 3 and 3 22 xxhxxf

complete the table below for each function and then graph each function on the axes provided

an invariant point is a point that remains unchanged when a transformation is applied to it

it is any x-intercept of a functions before the absolute value has been applied to the function

a piecewise function is a function composed of two or more separate functions or pieces, each with its own

specific domain, that combine to define the overall function

the absolute value function y = |x| can be defined as the piecewise function

0 if ,

0 if ,

xx

xxy

the absolute value symbol is found at MATH/NUM/1 abs( on the calculator

the zero of a function 0,x is the x-value where a line crosses the x-axis (we can use our calculators by

pressing 2nd/TRACE/2)

put your cursor on the zero, press your left hand arrow 3 times, press ENTER, move your cursor back to

the zero, press your right hand arrow 3 times, press ENTER/ENTER (never Guess?)

x f(x) g(x)

-2

-1

0

1

2

x f(x) g(x)

-2

-1

0

1

2

1. How can you create a graph of

an absolute value function

from a linear function?

2. What are the domain and range

of each function?

1. How can you create a graph of

an absolute value function from

a quadratic function?

2. What are the domain and range

of each function?

e.g. Consider the absolute value function 42 xy .

1. Sketch the graph of 42 xy on the axes to the right.

2. Sketch the graph of 42 xy by reflecting all the points below

the x-axis over the x-axis.

3. Determine the y-intercept and x-intercept.

4. State the domain and range of the function.

5. What is the invariant of the transformation?

6. Express as a piecewise function.

e.g. Consider the absolute value function 322 xxy .

1. Complete the square to convert to vertex form, khxay 2

.

Sketch the graph of 322 xxy on the axes to the right.

2. Sketch the graph of 322 xxy by reflecting all the points

below the x-axis over the x-axis.

3. Determine the y-intercept and x-intercept.

4. State the domain and range of the function.

5. What are the invariants of the transformation?

6. Express as a piecewise function.