Graphing Piecewise Functions
Course: Algebra II, Advanced Functions and Modeling
Materials: student computers with Geometer’s Sketchpad, Smart Board, worksheets (p. 3-7 of this document), colored pencils
Strategies: technology exploration
• The student will be able to graph functions using Geometer’s Sketchpad.
• The student will be able to select an appropriate straightedge option and use the display options to highlight particular sections of a function graph based on a given domain.
• The student will be able to apply these skills to graph piecewise functions with Geometer’s Sketchpad.
Introduction: The teacher will review the steps for graphing functions in Geometer’s Sketchpad by graphing each function in Part A of the worksheet. Next the teacher will introduce the term domain, and help students apply this to identify the portion of each graph that matches the given domain. The teacher will initially use the markers to highlight the specified sections, but once students understand the term domain, she will show them how to use the straightedge tools to highlight these portions of the graph using Geometer’s Sketchpad. The students will model this on their computers and use colored pencils to create similar graphs on their worksheets.
domain – the x-values of a function Geometer’s Sketchpad Steps:
1. In the Graph menu, select Show Grid.
2. Graph the given function (Graph, Plot New Function).
3. Use the point option to place points on the function at the boundaries of the domain. Remember that when the point is exactly on the line, the line will change color.
4. Select a straightedge option and using a thicker line, highlight the portion of the function that matches the domain. (Hint: If the function exists between two values, use the segment option. If the function continues to ±∞, use the ray option.)
5. You may opt to hide the original function by clicking on a portion of the function outside the domain and choosing Hide Function Plot from the Display menu. This will show only the darkened section based on the domain.
Lesson Development: The teacher will introduce the concept of piecewise functions and show students how the problems in Part A of the worksheet can be combined to form a piecewise function (Part B #4). A GSP document (Graphing Piecewise Functions.gsp) will be prepared ahead of time that has all three functions
graphed on the same coordinate plane with the appropriate portions highlighted. The teacher will show students what the graph looks like with the functions hidden so only the portions related to the domain of the piecewise function are visible. This will give students a good opportunity to see that piecewise functions may have disconnected portions and often appear to have distinct sections. The open or closed endpoints will be drawn on the graph using the Smart Board markers.
Guided Practice: Each student will complete Part B of the worksheet with a partner. When the teacher observes that most people have completed a particular problem, she will ask student to stop working for a few minutes to allow a student pair to present the graph on the Smart Board. This will allow students to check their work throughout the activity so they can catch any mistakes early in the practice activity.
Assessment: Informal assessment will be done throughout the class period. The teacher will determine whether or not students have met the stated objectives through their work and presentations during the guided practice section. Formal assessment will be based on the homework practice worksheet.
Geometer’s Sketchpad Hints
To show the coordinate plane: 1. Choose the Graph menu. 2. Select Show Grid. To graph a function: 1. Choose the Graph menu. 2. Select Plot New Function.
3. Type in the function you would like to graph. You do not need the y= portion of the equation.
* To type an absolute value function, select the Functions button and choose abs. Whatever appears inside the parentheses behind “abs” will be included inside the absolute value bars for the graph.
To darken a particular section of a function:
1. Select the point option on the left side of the screen.
2. Place points on the function at the domain boundaries. When the point is exactly on the line, the line will change color. You can also use Plot Points in the Graph menu to be more exact. If you are darkening a section of an absolute value graph that includes the vertex you will need points at the boundaries and an additional point at the vertex.
3. Select a straightedge option on the left side of the screen. If you want to darken an area of the graph between two values, you should use the segment option and click on each of the points at the boundaries. If you want to darken an area that extends to ±∞, you should use the ray option and click on the boundary point then another point in the domain specified.
4. If you would like to hide the remainder of the function, highlight the original function (and not the darkened section). From the Display menu, select Hide Function Plot.
Using Geometer’s Sketchpad, graph each function and darken the region associated with the domain. On the graph square given with the problem, sketch the function lightly with pencil and use a colored pencil to darken the region associated with the domain.
2. y =1 domain: x<−1
3. 4y =2x− domain: x>3
Using Geometer’s Sketchpad, graph each function, and darken the region associated with the domain. On the graph square given with the problem, sketch the function lightly with pencil and use a colored pencil to darken the region associated with the domain.
4. ⎪ ⎩ ⎪ ⎨ ⎧ > − ≤ ≤ − − < = 3 4 2 3 1 1 1 x x x x x y
5. ⎩ ⎨ ⎧ − > − − ≤ + = 2 2 4 x x x x y 6. ⎩ ⎨ ⎧ ≥ − < + − = 3 8 3 1 2 x x x x y 7. ⎩ ⎨ ⎧ − > + − − ≤ + = 4 3 4 2 x x x x y
Piecewise Functions Practice
Graph each piecewise function. You may use your calculator or Geometer’s Sketchpad to graph the individual functions, but they are all functions that you should be able to graph without a calculator.
1. ⎩ ⎨ ⎧ ≥ − < + − = 2 1 2 1 x x x y 2. ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ > + − ≤ < − − ≤ − − = 3 3 4 1 3 3 3 3 3 4 1 x x x x x x y
3. ⎪ ⎪ ⎪ ⎩ ⎪⎪ ⎪ ⎨ ⎧ < ≤ − < ≤ − < ≤ − − < ≤ − − < ≤ − = 5 3 6 3 1 3 1 1 0 1 3 3 3 5 6 x x x x x y 4. ⎪⎩ ⎪ ⎨ ⎧ ≥ + < = 3 2 3 1 3 2 x x x x y