Algebra II Honors—Day 10

Procedures

• Pick up the following from the table:– Handout, whiteboard, marker, eraser

• Get into groups of three or four students.

Goals for Today

• Reminder—First Graded Homework Assignment (checked for accuracy)—tomorrow—Tuesday, Sept. 10

• Quotable Puzzle due today• No homework check today• Essential Questions

– New Material/Group Investigations• Homework

Essential Questions

• How do the values of a, b, c, and d in the function affect the graph of ?xxf )(

dcbxaxf )(

Exploring the Absolute Value Function

• The absolute value function is a function of the form

• It has a two-part definition as follows: for

and for

xy

xy 0x

xy 0x

The Graph

For x≥0, the graph is the same as the line y=x (intercept of 0 and slope of POSITIVE 1)

For x<0, the graph is the same as the line y=-x (intercept of 0 and slope of NEGATIVE 1)

y=|x|

Explanations

• f(x)=a|bx+c|+d

The letters a, b, c, and d represent shifts or changes to the basic “parent” graph of f(x)=|x|. Each of these plays a different role in the movement of the graph. You will explore each.

Assignment• Within each group divide up the work in each section

on the handout and then compare answers for all the parts. Use the whiteboards first until all graphs are finished.

• Draw the basic graph: f(x)=|x|. This is called the “parent graph” or the “parent function.”

• Make a table of points for the other equations and graph each one on the same coordinate plane. For each graph, use the domain {-3, -1, 0, 1, 3} and find the y-values. Then draw the graph.

The effect of “a” (the number outside the function—multiplies the entire function)

xxf 3)(

The parent function

xxf2

1)(

xxf )(

New function “grows” faster• called a “vertical stretch”• slope is steeper—

multiplied by “a”

New function “grows” slower• called a “vertical shrink”• slope is less steep—

multiplied by “a”• the negative flips it upside

down as well

The effects of “b” and “c” (the numbers inside the function—“b” multiplies the x only)

3)( xxf

The parent function

5)( xxf

xxf )(

b=1 and c=3• New function is shifted

LEFT 3 units (“-c”) and slope is still the same as parent

b=1 and c=-5• New function is shifted

RIGHT 5 units (“-c”) and slope is still the same as parent

The effects of “b” and “c” (the numbers inside the function—“b” multiplies the x only)

36)( xxf

The parent function

93)( xxf

xxf )(

b=6 and c=3• New function is shifted

LEFT 1/2 unit (“-c/b”) and slope is multiplied by 6 (“b”)

b=3 and c=-9• New function is shifted

RIGHT 3 units (“-c/b”)and slope is multiplied by 3 (“b”)

The effects of “d” (the number added outside the function)

8)( xxf

The parent function

6)( xxf

xxf )(

d=8• New

function is shifted UP 8 units (“d”)

d=-6• New

function is shifted DOWN 6 units (“d”)

In your notes

• Make sure you can summarize the effects of each of the numbers a, b, c, and d in the equation . These numbers will be used throughout this course for other functions.

• Understand that each of these numbers either “shifts” or “stretches” the parent function.

dcbxaxf )(

Homework

• Absolute Value Graphs– Do without a calculator based on what you

learned.Problems 1, 2, 4, 7, 8, 9, 10, 12