Graphing Quadratic Functions in Vertex Form f(x)=a(x-h)^2 + k.

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Objective

SWBAT graph quadratic functions in Vertex Form by identifying the Vertex from the equation, and plotting 2 points on each side of the vertex.

Big Idea

The Parent Function is the focus of this lesson to identify transformations of every point on the graph by identifying the transformation of the Vertex.

Warm Up

10 minutes

I begin this lesson with a Warm Up on the second slide of this Power Point. I hand each student a copy of the Power Point to write on and use to graph functions as we move through the lesson today.  The Warm Up shows two graphs for the students to compare.  One is the Parent function, y equals x squared. The other function is the quantity of x minus one squared plus two.  I expect the students to be able to explain that the second function shifts one to the right and two up from the Parent function.  

As I am reviewing the Warm Up, I call on random students to explain how the two graphs are similar and how they are different. I expect students to say that the two functions are both Quadratic Functions in the shape of a Parabola, and then to focus on the transformation from the Parent Function to the other given function. 

I emphasize that the key point to focus on for the transformations of Quadratic Functions is the Vertex.  Then I ask the students, "Did all of the points on the Parent Function x squared shift one to the right and two up?" Some students are hesitant about what the answer is to this question.  So we check other points on the two Parabolas to verify that all of the points shift one to the right and two up.  

Again, I emphasize that the key point to focus on for the transformation of Quadratic Functions is the Vertex, but students should recognize that all of the other points on the graph make the same transformations.  

After students make the connection between the points on the graph and the transformation, we focus on the meaning of the formula.  I state to students that in this lesson, they are going to learn how a, h, and k affect the graph of a Quadratic Function.

I have students identify a, h, and k and then we discuss how it affects the graph.  Students understand that a has the same effect as it did in the Standard Form of a Quadratic Function. We discuss the transformations listed below:

  • If a is negative the graph reflects across the x-axis
  • If a is greater than one, it is a vertical stretch that makes the graph narrower.
  • If a is less than one, it is a vertical shrink that makes the graph wider
  • (h, k) is the Vertex of the Quadratic Function
  • h is the horizontal shift from the Parent Function
  • k is the vertical shift from the Parent Function

I have students get out their foldable that they made in the first lesson of this unit, and turn to Vertex Form.  On the foldable, students make a note that h is opposite of the sign in the parentheses for the horizontal shift, and k is the exact sign given.  This is what helps students identify the transformation of the Vertex.  It is confusing for the students at this level to focus on the formal definitions of h and k until they practice more.

 

Power Point

30 minutes

After reviewing the Warm Up, I continue working with students on slide three of the Power Point.  I model for students how to graph a Quadratic Function in Vertex Form.  Just like Standard Form, students should identify the Vertex first!

It is easier in Vertex Form because the Vertex can be identified directly from the equation. However students need to be careful of their signs.  If a student makes an error on the sign of either coordinate of the Vertex, the graph will be incorrect.

The next steps are the same as graphing a Quadratic Function in Standard Form.  After the Vertex is identified, it is placed in the middle of the t-table.  Students need to select one to two x-coordinates on each side of the Vertex. Then the corresponding y-values are found by substituting the x-coordinate into the function given.  The students need to focus on finding the y-coordinates on one side of the Vertex only. The other side of the Parabola can be plotted by the students knowledge of the Parabola being symmetric across the axis of symmetry.  I demonstrate reviewing slide three with students in the video below.

 

After reviewing slide three with the students, I assign students to continue graphing the Quadratic Functions given on slide four through slide seven shown below:

I walk around to monitor student progress and assist students as they are working on graphing the remainder of the Quadratic Functions.

Finally, I have students stand behind their chairs for a Kinesthetic Activity on slide eight. Students are to stand in rows either behind their desk, or at the back of the room if there is more space.  I have two student volunteers lead the students in the movements of the transformations from the Parent Function. The two volunteers face the screen, and have their backs to the other students. In this way, all student movements are in the same direction. 

Students have fun stepping out the movements of the transformations of each function posted on the screen. Students step two to the right and one up (forward) for the first given function, the quantity of x minus two squared plus one.  The students hold their arms up in the u-shape as they step to the movements.  For the third function, students flip their arms downward in the shape of a u for the function y equals negative x squared.  Students make their arms wider for a equal to 1/4 in number five, and narrower for a equal to three in number six.  

 

Exit Slip

10 minutes

With about 10 minutes remaining in the period, I post the Exit Slip which is also on slide nine of the Power Point on the board.  Students already have a copy of the Exit Slip in the Power Point packet that I handed to them at the beginning of class today. Students are to hand in the Exit Slip before leaving class.  

If students did not complete graphing all of the functions on slides four through seven, they are to complete it for homework.  Most of the students were able to explain the differences in the two given functions on the Exit Slip.

Function one on the Exit Slip had the following transformations from the Parent Function:

a.  one to the right

b.  three up

c.  narrower (Vertical Stretch)

Function two on the Exit Slip had the following transformations from the Parent Function

a.  three to the right

b.  two up

c.  wider (Vertical Shrink)