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Aim: Identifying the graph of different parabola.
Material: (1) Copy of 2003 Regents question
(2) Navigator
(3) Class set of T-83 Plus/T-84
(4) Handouts
Mini Lesson: Todays lesson will focus on determining the functions/equations of different parabolas given the respective graph of the parabola. This was a question that was asked on the New York State Regents January 2003 (Question 29). What we want is to gradually develop the characteristics of a parabola so that the students will by the end of the lesson know how to develop the function of any parabola, given any graph. It can take two lessons depending on how fast you wish to move along. This lesson does not view a parabola in the standard form of a quadratic formula.
The lesson will begin by displaying question number 29, through the Navigator and then allowing the students to have one guess at the function rule of the parabola.
Step 1: After the students had the opportunity to guess at the equation of the parabola, we begin by graphing the general formula of the parabola, y1 = x2. We go into APPS and pull up TRANSFRM. After we get the TRANSFORMATION window we hit enter. You should get a clear screen. When you go to y = , you should notice that the far left side of y is not the normal line but what appears to be a hour glass on its side. Key in y2 = Ax2 + B, and consider what happens when we change the values A and B in y2 = Ax2 + B.
Step 2: Then we will ask the students what they expect the graph to look like if fix A at A = 1 and B = 2 in the function y = Ax2 + B, that is y = x2 + 2.
Note: Though we have two equations or more equations when we are in the TRANFORMATION mode only one graph will show up on the screen.
Step 3: Students will realize that as we change the B value the parabola will move/ride up and down along the y-axis. As we continue to change the values of B only, the students will identify that the parabola will move up is B is positive and down is B is negative.
Step 4: The next question we will consider is what would happen if I wanted to move the parabola to the right or to the left. So first lets look at two to the right of the origin, how would I modify the function rule? This time we key in y = (Ax - B)2+ C where A = 1, B is allowed to vary, and C = 0. Students in exploring this will identify what happens to the parabola. That is, the parabola will move/ride along the x-axis, in lieu, of the y- axis based on the value of B.
Step 5: The next step will be to have the students identify what we can do to move the parabola out to the right and up away from the x, and y axis. The students based on their exploration will have realized from Step 1, and Step 2 that we can move the parabola about the plane by changing the values of B and C in the functions. Continuing with our function y =
(Ax - B)2 + C, if we change the value B the parabola will move to the right or to the left (based on the sign of the number). If we change the value of C the parabola will move up or down. One common fact we should remind or ask our students is when we move the parabola to the left for the function y = (Ax - B) 2 + C, when B is negative the value of - B is positive. Therefore, looking at the graph, if we hold the value of A fixed at 1, we have the following examples:
Notice as the value of B and C for the displacement of the parabola.
Step 6: Here we reintroduce the original equation of the regents in order to assess our students. If they get it incorrect reassess by discussing the concept they should have written in there graph draft (handout).
As a further assessment the students will then be asked suppose we had the equation y = x3, what would we have to do to the function so that the turning point/point of inflection will lie on the coordinate (x, y) = (2, 3). The approach to this question may be asked two way, either what is the new function rule, or where would we expect our graph top appear? Our function rule will change from y = x3 to y = (x-2) 3 + 3
Other issues that should follow-up on this lesson if not already done in a prior lesson is what happens when we change the coefficient of the value A in any of the above steps makes the parabola either open wider or contract in. Also, another matter that needs to be considered if not done so already is if the coefficient is negative the parabola will have a maximum value as oppose to a minimum value.
If time allows do the following fun activity:
Step 7: Students will be told: Suppose you are hired as a contractor to build a bridge for $1,000,000. We wanted to build a bridge that has one underpass on each side of the y axis. And the bridge is as high as y =5. The parabolas vertex sits on the coordinate negative (-6, 4) and (6, 4). I also want two cars moving along the road after the bridge is done. We may have as one possibility y1 = 5, y2 = -.2(x-6)2 + 4, y2 = -.2(x + 6)2 + 4. Notice I adjusted the wide of the parabola by multiplying it by -.2. Also in order so that our car is slightly above the road y=5 I keyed in the value y= 5.2, and changed the graph feature to a circle for y4 and y5. Note below for the steps and the graph of the bridge.
Of course this can continue with two underpasses on each side of the y-axis. Stating suppose we get a call from the contractors and they say they now are willing to pay an additional
$500,000 for a bridge with four under-paths under the bridge. The question I have for you is this Deal or no deal?
After the bridge is done all the students should sent their cars through Navigator over the bridge
Homework:
(1) Students will manipulate the equations for each of the previous examples that were done in class so it will be in a standard quadratic equation, and identify what properties of the standard quadratic allows the parabola to move about the plane. That is, what properties of the standard equation help them identify the properties that makes the parabola move up, down, to the right, to the left, or any combination?
(2) Does this concept hold true for any other type of function, say:
(a) y = 1/x,
(b) y = 2x
(3) Based on what you learned about the movement of the parabola, what can you say about
y = (x +2) 1 (remember this is a linear equation), or where on the coordinate plan would you expect this to be?
Miguel (Mike) A. Vazquez Texas Instrument T-83 Plus
Region 1/District 9 mvazque5@nycboe.net
Lesson Plan
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