Do Now

Make a table for –2 ≤ x ≤ 2 and draw the graph of:

y = 2x+1 + 3

(Problem #1 from today’s packet)

6

4

2

-2

-4

-6

-10 -5 5 10

Absolute Value Functions

Absolute value functions are functions that contain absolute value in their equations.

The most basic absolute value function is:

y = |x|

First We’ll Graph a Similar Equation…The line y = x

6

4

2

-2

-4

-6

-5 5

Let’s Compare Tables

x y

-3 -3

-2 -2

-1 -1

0 0

1 1

2 2

3 3

x y = |x| y

-3

-2

-1

0

1

2

3

y = x y = |x|

6

4

2

-2

-4

-6

-5 5

Let’s See What Happens On The Graph

Adding a Coefficient Inside

x y

-3 3

-2 2

-1 1

0 0

1 1

2 2

3 3

x y = |2x| y

-3

-2

-1

0

1

2

3

y = |x| y = |2x|

6

4

2

-2

-4

-6

-5 5

The graph gets steeper on both sides and closer to its line of symmetry.

In Your Calculator

Let’s look at:

y = |4x|

y = | x |2

1

y = |– 2x|

x y

-3 6

-2 4

-1 2

0 0

1 2

2 4

3 6

x y = |– 2x| y

-3

-2

-1

0

1

2

3

y = |2x| y = |– 2x|

The graph of the absolute value function takes the shape of a V.

What are some observations that you can make about these graphs?

1.  As the coefficient of x gets larger, the graph becomes steeper, closer to its line of symmetry.

2.  As the coefficient of  x gets smaller, the graph becomes less steep, further from its line of symmetry.3.  If the coefficient of  x is negative, the graph is the

same as if that coefficient were positive. Absolute value changes negative values to positive values. 

Notice that the graphs of these absolute value functions are on or above the x-axis.  Absolute

value always yields answers which are positive or zero.

In Your Calculator

Let’s look at:

y = |x| + 3

y = |x| – 5

Challenge

See if you can shift the graph right or left by adding or subtracting a number from somewhere else in the equation.

y = |x + 3|

y = |x – 5|

Summary

Adding or subtracting a number outside the absolute value shifts the graph up or down.

Adding or subtracting a number inside the absolute value shifts the graph right or left.