Name: ____________________________________ Date



Name: ____________________________________ Date: ______________________________

Absolute Value Transformations - Notes

Vocab:

• Absolute value is the measure of the distance away from zero on a number line.

▪ Since absolute value is the measure of distance it can never be negative!

• Parent Function is the simplest form of a family of functions

▪ Absolute value parent function is y = │x│

• Transformation is a change made to a function (in the equation or graph)

• Vertex is the point where the graph changes directions

Let’s make a table and graph the absolute value parent function.

|x |Y |

|-5 | |

|-4 | |

|-3 | |

|-2 | |

|-1 | |

|0 | |

|1 | |

|2 | |

|3 | |

|4 | |

|5 | |

What do you notice about the graph?

Vertex: The point where the graph changes directions.

What is the vertex of the absolute value parent function? How can you tell from the table and graph?

Since we know not all equations are the same, lets explore what happens if we change the parent function.

1.) Using the graphing calculator:

a. click on apps

b. Scroll down to Transfrm (or hit alpha T (4))

c. Hit enter and then any key to go back to the home screen

Now let’s tell the calculator what we want to transform!

a. go to y = (notice the pause sign….this means the transform app is on)

b. type abs(x) + A {abs can be found in the MATH, NUM)

c. hit graph

d. Use the right and left arrows to see how the graph changes when we add or subtract a value.

QUESTION: What do you notice?

QUESTION: Write a rule to explain what happens to the absolute value graph when we add or subtract a value outside the absolute value sign.

QUESTION: Pick three A values and write the equation, sketch the graph, identify the vertex, and describe the transformations:

Equation: _______________ Equation: _______________ Equation: _______________

Vertex: _________________ Vertex: ________________ Vertex: _________________

Transformation: Transformation: Transformation:

2.) Now let’s explore what happens when we add or subtract a value inside the absolute value sign.

a. go to y = (notice the pause sign….this means the transform app is on)

b. type abs(x + A)

c. hit graph

d. Use the right and left arrows to see how the graph changes when we add or subtract a value.

QUESTION: What do you notice?

QUESTION: Write a rule to explain what happens to the absolute value graph when we add or subtract a value inside the absolute value sign.

QUESTION: Pick three A values and write the equation, sketch the graph, identify the vertex, and describe the transformations:

Equation: _______________ Equation: _______________ Equation: _______________

Vertex: _________________ Vertex: ________________ Vertex: _________________

Transformation: Transformation: Transformation:

3.) Now let’s explore what happens when multiply the absolute value by a coefficient.

a. go to y = (notice the pause sign….this means the transform app is on)

b. type A abs(x)

c. hit graph

e. Use the right and left arrows to see how the graph changes when we multiply by a coefficient

QUESTION: What do you notice?

QUESTION: Write a rule to explain what happens to the absolute value graph when we multiply the absolute value parent function by a coefficient

QUESTION: What happens to the graph when ‘A’ becomes negative? Write a rule to describe the effects of a negative a value.

REFLECTION OVER X-AXIS: ___________________________

QUESTION: Pick three A values and write the equation, sketch the graph, identify the vertex, and describe the transformations:

Equation: _______________ Equation: _______________ Equation: _______________

Vertex: _________________ Vertex: ________________ Vertex: _________________

Transformation: Transformation: Transformation:

Putting it all together:

The general form of an absolute value function is:

y = a │x + h│+ k

Identify the effects of each parameter:

A:

H:

K:

Example: Describe the transformations of each absolute value equation:

a.) y = │x – 4│ b.) y = ½│x│ c.) y = │x + 1│ - 2 d.) y = -2│x│

Example: Write the equation of each absolute function that has the following transformations:

a.) Vertex is (4,1) b.) vertex is (-2,3) and has a vertical dilation and a stretch by 3

c. vertex (-3, 0) and reflected over x axis d.) vertex is (-1, -4) with a reflection over x axis

e.) vertex at origin, vertical dilation by 2/3

Now let’s graph an absolute value function without using the graphing calculator.

1. Start by writing down all the transformations

2. Move your translations (plot your vertex)

3. Apply and vertical dilations and reflections

4. Connect the points to make a smooth curve

Example: Graph the absolute value equation y = -2│x – 4│ + 3

Transformations: Graph:

We can answer all our function questions using our graph!

a. Is this a function? Explain:

b. What is the domain:

c. What is the range:

d. What is the x intercept: e.) What is the y intercept:

f.) Is there a max or min? g.) Vertex:

If so, what is it?

Where is it?

h. Describe the end behavior:

a. As x ( ________, y ( _______

b. As x ( ________, y ( _______

i. What are the zero(s)?

j. Interval increasing: k.) Interval decreasing:

You try: Graph each absolute value function and answer all the function questions!

a.) f(x) = ½│x – 3│ - 1 b.) y = -3│x + 4│ c.) f(x) = -2/3│x│ + 4

Transformations: Transformations: Transformations:

Vertex: Vertex: Vertex:

|Question |Graph a |Graph b |Graph c |

|Is it a function? Explain. | | | |

|Domain: | | | |

|Range: | | | |

|X intercept: | | | |

|Y intercept: | | | |

|Zeros: | | | |

|Interval increasing: | | | |

|Interval decreasing: | | | |

|Max/Min: | | | |

|Vertex: | | | |

|End Behavior: | | | |

|As x → ∞, y → | | | |

|As x → -∞, y → | | | |

|Transformations | | | |

| | | | |

| | | | |

Now let’s reverse the process! For each graph, describe all transformations and write the equation!

a.) b.)

a.

Transformations: Transformations:

Vertex: Vertex:

Equation: Equation:

c.) d.)

Transformations: Transformations:

Vertex: Vertex:

Equation: Equation:

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