Math II Unit 5 (Part 1)

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

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

 Symbol lxl  The distance x is from 0 on the number line.  Always positive  Ex: l-3l=

 What are the possible values of x? x = 5 or x = -5  Another example… |x|= 13  What are the possible values of x?

 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!

|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.

6x - 3 = 15 or 6x - 3 = -15 6x = 18 or 6x = -12 x = 3 or x = -2 * Plug in answers to check your solutions!

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 = -18 x = 2 or x = -9 Check the solutions.

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

How do absolute value equations and absolute value inequalities differ?

1. |ax+b| 0 Becomes an “and” problem Changes to: –c < ax+b < c 2. |ax+b| > c, where c > 0 Becomes an “or” problem Changes to: ax+b > c or ax+b < -c

 Becomes an “and” problem

 Get absolute value by itself first.  Becomes an “or” problem

How do absolute value equations and absolute value inequalities differ?

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.

Take your card (either the graph or the interval notation) to the person who has your “match.” You are finding your TWIN.

 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.

 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.

 Each equation corresponds to a different part of the domain. Find 1. f(-1)2. f(0)3. f(5)

EQ: How can piecewise functions be described?

 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)

 Give the characteristics of the function.  Domain:  Range:  X-intercepts:  Y-intercepts:  Intervals of increase/decrease/ constant:  Extrema:

EQ: How do I identify points of discontinuity?

 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.

 The “breaks” or “holes” are called points of discontinuity.  This graph has a point of discontinuity where x = 2.

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.

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.

 A step function is an example of a piecewise function.  Let’s graph this example together.

 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.

EQ: How are graphs of step functions used in everyday life?