1.7 Piecewise Functions Students will be able to Identify and Graph piecewise functions (including absolute value) as evidenced by a math journal.

What is a piecewise function? Students will be able to Identify and Graph piecewise functions (including absolute value) as evidenced by a math journal. What is a piecewise function? A piecewise function (passes vertical line test) is a function that is made up of different graphs for different intervals. “Frankenstein Functions”

What common functions are used to “piece” together? Students will be able to Identify and Graph piecewise functions (including absolute value) as evidenced by a math journal. What common functions are used to “piece” together? Constant Just a number, looks like a horizontal line. Need at least two points Linear 𝑚𝒙+𝑏, what is the slope? What is the y intercept? Need at least 2 points Quadratic 𝑎 𝑥 2 +𝑏𝑥+𝑐 or 𝑎 𝑥−ℎ 2 +𝑘 whose vertex is (ℎ, 𝑘) and 𝑎 determines directions of opening (up or down) and compression (wide or skinny) Need at least three points Cubic 𝑎 𝑥 3 where 𝑎 is the directions of opening and compression, need three points

How do we graph piecewise functions? Students will be able to Identify and Graph piecewise functions (including absolute value) as evidenced by a math journal. How do we graph piecewise functions? Use multiple colors if you have them, one color for each piece. Know the difference between the symbols <> means NOT included so we use and OPEN circle ≤≥ means included, so we use a CLOSED circle Understand what the boundaries mean, four types 𝑥>𝑎 all values to the right of a 𝑎<𝑥<𝑏 all values between a and b 𝑥<𝑏 all values to the left of b 𝑥=𝑐 only plot a single point Determine WHERE (boundaries) you are graphing WHICH functions (colors help here!!) Create a table for each piece, your table must include the boundaries AND at least one other point. Determine if boundaries are open or closed and graph. USE COLOR!!

Example 1: Graph 𝑓 𝑥 = 1, 𝑥≤−2 𝑥+2, −2<𝑥≤ 3 2𝑥 𝑥>3 Students will be able to Identify and Graph piecewise functions (including absolute value) as evidenced by a math journal. Example 1: Graph 𝑓 𝑥 = 1, 𝑥≤−2 𝑥+2, −2<𝑥≤ 3 2𝑥 𝑥>3 Since we have 3 pieces we will need 3 t-tables (make sure to include the boundaries) Do our tables have the boundaries? Are the boundaries open or closed? 𝒙 𝒇 𝒙 =𝟏 −𝟐 −3 𝒙 𝒇 𝒙 =𝟏 −𝟐 1 −3 𝒙 𝒇 𝒙 =𝟏 𝒙 𝒇 𝒙 =𝒙+𝟐 −𝟐 𝟑 5 𝒙 𝒇 𝒙 =𝒙+𝟐 −𝟐 𝟑 𝒙 𝒇 𝒙 =𝒙+𝟐 𝒙 𝒇 𝒙 =𝟐𝒙 3 4 𝒙 𝒇 𝒙 =𝟐𝒙 𝒙 𝒇 𝒙 =𝟐𝒙 3 6 4 8 CLOSED OPEN OPEN CLOSED

Example 2: Graph 𝑓 𝑥 = 𝑥+2 2 −3 𝑥<−1 𝑥 3 −1≤𝑥≤1 −𝑥 2 𝑥>1 Students will be able to Identify and Graph piecewise functions (including absolute value) as evidenced by a math journal. Example 2: Graph 𝑓 𝑥 = 𝑥+2 2 −3 𝑥<−1 𝑥 3 −1≤𝑥≤1 −𝑥 2 𝑥>1 𝒙 𝒇 𝒙 = 𝒙+𝟐 𝟐 −𝟑 𝒙 𝒇 𝒙 = 𝒙+𝟐 𝟐 −𝟑 −𝟏 −2 −3 𝒙 𝒇 𝒙 = 𝒙+𝟐 𝟐 −𝟑 −𝟏 −2 −3 𝒙 𝒇 𝒙 = 𝒙 𝟑 −𝟏 −1 𝟏 1 𝒙 𝒇 𝒙 = 𝒙 𝟑 −𝟏 𝟏 𝒙 𝒇 𝒙 = 𝒙 𝟑 𝒙 𝒇 𝒙 =− 𝒙 𝟐 𝟏 −1 2 −4 3 −9 𝒙 𝒇 𝒙 =− 𝒙 𝟐 𝒙 𝒇 𝒙 =− 𝒙 𝟐 𝟏 2 3 OPEN CLOSED OPEN CLOSED Determine which function would be used to find each value: 𝑓 1.5 𝑓 −.7 𝑓 32 𝑓 −1 𝑓 −1.1

Students will be able to Identify and Graph piecewise functions (including absolute value) as evidenced by a math journal. What is absolute value? Absolute value || is the SIZE or a number (one, two, three dimensional). In one dimension we usually say “|−3| means how far is −3 away from zero?” and we say 3, so |−3|=3. As we progress through this course we will see what || means in higher dimensions. 𝑥 means −(𝑥)AND 𝑥 since they are both the same distance from zero. 2 𝑥 +2 then becomes 2∗−(𝑥) +2=−2𝑥+2 AND 2 𝑥 +2=2𝑥+2 |3𝑥−4| becomes − 3𝑥−4 =−3𝑥+4 AND 3𝑥−4 =3𝑥−4

How do I graph absolute value? Students will be able to Identify and Graph piecewise functions (including absolute value) as evidenced by a math journal. How do I graph absolute value? Create a t-table and start plotting points, your goal is to see that absolute values “v”, since you are piecing together two linear functions, once you find where the sharp turn happens you can use a ruler to continue the line. The more comfortable you are graphing absolute value, the fewer numbers you will need to plot For our purposes in THIS SECTION, we will be working with LINEAR absolute value. We will see later that we can have quadratic, cubic, etc absolute value also.

Example 3: Graph 𝑓 𝑥 =2 𝑥 +2 Create a t-table Students will be able to Identify and Graph piecewise functions (including absolute value) as evidenced by a math journal. Example 3: Graph 𝑓 𝑥 =2 𝑥 +2 Create a t-table Plot the points as you go until you see the “v” 𝒙 𝟐 𝒙 +𝟐 What are the two linear equations that would be the pieces? What would the boundary be? 𝑓 𝑥 = 2 𝑥 +2 2 −𝑥 +2 𝑓 𝑥 = 2𝑥+2 𝑥≥0 −2𝑥+2 𝑥<0

Example 4: Graph 𝑓 𝑥 = 3𝑥−4 𝒙 𝟑𝒙−𝟒 3𝑥−4=0 3𝑥=4 𝑥= 4 3 Students will be able to Identify and Graph piecewise functions (including absolute value) as evidenced by a math journal. Example 4: Graph 𝑓 𝑥 = 3𝑥−4 𝒙 𝟑𝒙−𝟒 3𝑥−4=0 3𝑥=4 𝑥= 4 3 What are the two linear equations that would be the pieces? What would the boundary be? 𝑓 𝑥 = +(3𝑥−4) −(3𝑥−4) 𝑓 𝑥 = 3𝑥−4 𝑥≥4/3 −3𝑥+4 𝑥<4/3

Summary In a complete sentence explain the difference between the symbols <> and ≤≥. Describe the process of graphing piecewise functions to your friend who was absent today in student friendly language.