Graphing Quadratic Equations (Day 1 of 2)

Students will use the zero product property to complete the Do-Now in 5 minutes. Students may need to be probed to refer to their notes from our last class, as this topic was recently taught. Next, I will pick one student to come up to the front of the room to lead through the Do-Now review.

A student volunteer will then read our objective, "SWBAT graph quadratic functions on a coordinate plane."

Before we begin I will pass graded exit cards back to students, and I we will quickly review their responses.

During our last class we sketched quadratics using the zero product property and solving for its roots. Today we will fine-tune this process by calculating the additional points that will be on each graph.

Students will follow along using this Presentation and Guided Notes. I will guide students through each example using deliberate questioning:

• Last class we sketched quadratic functions and today we are going to graph quadratic functions. Is there a difference between a sketch and a graph?  
• If we graphic a quadratic using its roots, how many verifiable points will we have on each graph? 
• Why will we only have one or two?
• What do I mean what I say verifiable? How do we verify coordinates? When else have we verified coordinates during this school year?
• In order to have a more accurate view of this function we need to find more coordinates... We can easily find the x-intercept of a quadratic function by solving for its roots. What about the y-intercept? 
• What do you know about the y-intercept of a linear function?

I will then display this graphing calculator on the board. I will grab these four functions one at a time, make observations about the y-intercept of each function ONLY: y= x² + 5x + 4;  y= x² + 7x + 6;  y= x² + 3x - 54;  y= x² + 1. I will then continue to push student's thinking with more probing questions:

• What happen when we graph each quadratic function? Did we recognize any patterns?
• What conclusion can I make about the Y end of a quadratic function?
• Why does this work? Use a quick table to verify this response. 
• We now have the x and y intercepts of a quadratic function. 
• Are there any other important points that we should find?
• What do you know about the vertex of a quadratic function? How can you use the roots of a quadratic function to find information about its vertex?
• If you use this thinking to find the x-value of the vertex (or the point in the middle of the two roots) is there a way for me to find what the y-value is at that point? 
• Where have we seen this process at before?
• We found the location of the vertex by looking at the point that was in the middle of the two roots. Even though this works, there is a better way to do this. Let's examine some more quadratic functions and see if we can spot any patterns to help us.
• Let's look at the graph of y= x² + 4x + 3 on our online graphing calculator. Where are the x-intercepts? Where is the vertex? 
• Let's take a closer look at the three points that we are know...Do these three points (-3,0) (-1, 0) and (-2, 0) have any relationship with each other?
• Let's look at the graph of y= x² - 10x + 9 on our online graphing calculator. Where are the x-intercepts? Where is the vertex? 
• Let's take a closer look at the three points that we are know...Do these three points (-3,0) (-1, 0) and (-2, 0) have any relationship with each other? (1, 0) (9, 0) and (5, -16).
• If students are having trouble generating an algorithm for the vertex, at this point I will ask the class if they are able to make an addition sentences using just the x-values. 
• To find the x coordinate of the vertex, we can divide the sum of the roots by 2 
• (a + b)/2. 
• If we don't know the roots, we can use the a,b, and c values from the standard form of a quadratic function.
• -b/2a
• To find the y-coordinate of the vertex we can evaluate the function with the x-value of the vertex, just like with linear functions.
• Do we have any lines of symmetry in this function?  Can we the definition of symmetry to aid us in this process?
• Are there any other points we can find in this quadratic function? How?
• Are there any flaws in this method? What do you think?
• Let's go back and review the entire process. Does it matter where you start?

Students will work individually or in pairs to practice graphing quadratics using this Kuta Software handout. Students can graph each function directly on the paper, but some may choose to use the quadratic graph paper to help organize their thinking.

After students have completed the handout, I will print out the last page of the activity in order to check their own answers at a seat with a neighbor. After reviewing the answer key students should explain any errors that they may have made using a sentence next to each graph.

While the class is completing this activity, I will use this time to pull a small group of students to review factoring and the zero product property.

To close today's lesson I will first ask students to summarize the process used when graphing a quadratic function. Students will verbalize the steps that they take for the whole class to hear.

I will then ask the class to and turn and talk with a neighbor to discuss whether or not a quadratic function will always have at least one root and/or a y intercept. Pairs will be asked to justify their response to the group with a sketch or table on the board. Students will then complete an Exit Card.