MATH II CO-TEACHER TRAINING

Math II Unit 5 (Part 1)

Absolute Value Equations and Inequalities

MM2A1cSolve absolute value equations and inequalities analytically, graphically and by using appropriate technology.

EQ #1Can an absolute value equation ever have “no solution”?

Absolute Value Recap

Symbol lxl The distance x is from 0 on the number

line. Always positive Ex: l-3l= 3

-4 -3 -2 -1 0 1 2

Ex: |x | = 5

What are the possible values of x?

x = 5 or x = -5

Another example… |x|= 13 What are the possible values of x?

Now, let’s think about this…

What will happen if …│x + 6│= 16? What will the two equations be? What are the solutions? Another example… │x │= -8 … What are the solutions? Remember…Absolute value equations

cannot be equal to a negative number!

Solving Absolute Value Equations

Solving Absolute Value Equations Math II – Unit 5

Make sure the Absolute Value expression is isolated.

Set up two equations to solve.

Equation 1

Equation 2

X =

X =

Check your solutions…

Equation 1

Equation 2

To solve an absolute value equation:

|ax+b | = c, where c > 0

To solve, set up 2 new equations, then solve each equation.

ax+b = c or ax+b = -c

** make sure the absolute value is by itself before you split to solve.

Ex: Solve |6x - 3| = 15

6x - 3 = 15 or 6x - 3 = -156x = 18 or 6x = -12

x = 3 or x = -2

* Plug in answers to check your solutions!

Ex: Solve |2x + 7| -3 = 8

Get the absolute value part by itself first!|2x+7| = 11

Now split into 2 parts.2x+7 = 11 or 2x+7 = -11

2x = 4 or 2x = -18x = 2 or x = -9

Check the solutions.

Revisit EQ #1Can an absolute value equation ever have “no solution”?

EQ #2How do absolute value equations and absolute value inequalities differ?

Solving Absolute Value Inequalities

How do you solve absolute value inequalities?

│ax + b│< c

│ax + b│≤ c

│ax + b│> c

│ax + b│≥ c

Rewrite as

and solve f or x.

Rewrite as

and solve f or x.

Rewrite as

and solve f or x.

Rewrite as

and solve f or x.

Solve: │4 – 2x│ < 22 Solve: │2x – 5 │ > 19

Graphic Organizer by Dale Graham and Linda Meyer Thomas County Central High School; Thomasville GA

Solving Absolute Value Inequalities

1. |ax+b| < c, where c > 0Becomes an “and” problemChanges to: –c < ax+b < c

2. |ax+b| > c, where c > 0Becomes an “or” problem

Changes to: ax+b > c or ax+b < -c

Solve & graph.

Becomes an “and” problem

2194 x

219421 x30412 x

2

153 x

-3 7 8

Solve & graph.

Get absolute value by itself first.

Becomes an “or” problem.

11323 x

823 x

823or 823 xx63or 103 xx

2or 3

10 xx

-2 3 4

Revisit EQ #2How do absolute value equations and absolute value inequalities differ?

Piecewise Functions

INTERVAL NOTATION

Interval Notation Graphic Organizer

Interval notation is another method for writing domain and range.Symbols you need to know…•Open parentheses ( )- means NOT equal to or does not contain that point or value•Closed parentheses [ ] – mean equal to or contains that point or value•Infinity ∞ - if the graph goes forever to the right (domain) or forever up (range)•Negative Infinity −∞ - If the graph goes forever to the left (domain) or forever down (range)•Union Sign ⋃ - means joined together … this part AND this part

Use the open parentheses ( ) if the value is not included in the graph. (i.e. the graph is undefined at that point... there's a hole or asymptote, or a jump)

Use the brackets [ ] if the value is part of the graph or contains that point.

Parentheses Brackets

Whenever there is a break in the graph, write the

interval up to the point. Then write

another interval for the section of the graph after that part. Put a

union sign between each

interval to "join" them together.

FIND YOUR FAMILYTake your card (either the graph or the interval notation) to the person

who has your “match.”You are finding your TWIN.

Examples…Give the Domain

[0, ∞)

[-3] U [-2] U [-1, ∞)(-4, 4]

[-1] U [3, ∞)

Definition of a Piecewise Function

Piecewise functions are functions that can be represented by more than one equation (a function made with many “PIECES.”

Piecewise functions do not always have to be line segments. The “pieces” could be pieces of any type of graph.

This type of function is often used to represent real-life problems.

Definition of a Piecewise Function

Piecewise functions are functions that can be represented by more than one equation (a function made with many “PIECES.”

Piecewise functions do not always have to be line segments. The “pieces” could be pieces of any type of graph.

Evaluating Piecewise Functions

Each equation corresponds to a different part of the domain.

Find 1. f(-1) 2. f(0) 3. f(5)

CHARACTERISTICS OF A PIECEWISE GRAPH

EQ: How can piecewise functions be described?

Characteristics of Piecewise Graphs

Domain – x-values Range – y-values X-intercepts (zeros) – points where graph

crosses x-axis Y-intercept – point where graph crosses y-axis Intervals of Increase/Decrease/Constant –

read from left to right ALWAYS give x-values only write in interval notation

Extrema – Maximum (highest y-value of function) Minimum – (lowest y-value of function)

Example…

Give the characteristics of the function.

Domain: Range: X-intercepts: Y-intercepts: Intervals of

increase/decrease/constant:

Extrema:

POINTS OF DISCONTINUITY

EQ:How do I identify points of

discontinuity?

Continuous Function

Notice that in this case the graph of the piecewise function is one continuous set of points because the individual graphs of each of the three pieces of the function connect.

This is not true of all cases. The graph of a piecewise function may have a break or a gap where the pieces do not meet.

Discontinuous Function

This is not true of all cases. The graph of a piecewise function may have a break or a gap where the pieces do not meet.

The “breaks” or “holes” are called points of discontinuity.

This graph has a point of discontinuity where x = 2.

PARENT FUNCTIONS REVIEW

EQ:What are the six Parent Functions

from Math I and what are the characteristics of their graph?

Cut and Paste Activity: Have students match parent function properties to its name on graphic organizer.

Parent Functions Review

Domain of Function

Domain of Function

Domain of Function

Domain of Function

Domain of Function

Domain of Function

Name of Function

Name of Function

Name of Function

Name of Function

Name of Function

Name of Function

Range of Function

Range of Function

Range of Function

Range of Function

Range of Function

Range of Function

Intervals of Increase/Decrease

Intervals of Increase/Decrease

Intervals of Increase/Decrease

Intervals of Increase/Decrease

Intervals of Increase/Decrease

Intervals of Increase/Decrease

Parent Functions Review

GRAPHING PIECEWISE FUNCTIONS

1. Graph the function using parent graphs and transformations.

2. Use domain of function to find

"endpoints" of graph. Do this by substituting

in the x-values and finding the y-values.

(x, y)

3.Plot "endpoints" found in step #2

(Open circles if NOT included; CLOSED circles

if included) These points should lie on

your graph.

4. Erase function not located between

"endpoints."If only bounded on one

side (one endpoint) then the other endpoint is positive OR negative

infinity.

GRAPHING A STEP FUNCTION

Step Function

A step function is an example of a piecewise function.

Let’s graph this example together.

Step Functions

Ceiling Functions

In a ceiling function, all non-integers are rounded up to the nearest integer.

An example of a ceiling function is when a phone service company charges by the number of minutes used and always rounds up to the nearest integer of minutes.

Floor Functions

In a floor function, all non-integers are rounded down to the nearest integer.

The way we usually count our age is an example of a floor function since we round our age down to the nearest year and do not add a year to our age until we have passed our birthday.

The floor function is the same thing as the greatest integer function .

STEP FUNCTIONSEQ:

How are graphs of step functions used in everyday life?