Today you will be doing several explorations involving parabolas / quadratic equations.  These 4 explorations are intended to give you a more intuitive sense of why parabolas look the way they do (what effect the coefficients have on the shape of the parabola).  I hope you will find these visualizations helpful.

Please COPY AND PASTE the text of this web page into an email (or word doc) and answer the questions in EACH SECTION.  At the end of class, email your work to Ms. Krishnan.

Part I: Quadratic Practice

Open up the Quadratic Practice demonstration by clicking on "download live version".  (If it asks you what application to use, choose Mathematica Player.  If it asks you for permissions, allow them.  If it gives you warnings, just say ok.)

You should see 3 big dots and a parabola.  The 3 sliders control the 3 coefficients of the equation of the parabola (in standard form).  Your goal is to change the coefficients (by sliding the sliders) until the parabola goes through all 3 dots.  (They will turn red and it will say "Congrats!")  Do this at least 10 times, noticing how the parabola changes as you change each coefficient.

Question 1: Describe how changing each coefficient affects the parabola.  (Does it move?  Where/how?  Does it change shape?  How?)
(a)
(b)
(c)

Question 2: Which coefficient do you have to change to make the parabola point down instead of up?  Explain your answer.

Question 3: Do you think there could be more than one parabola that goes through the same set of 3 points?  (In other words, is the parabola that goes through 3 points the only one?)  Explain your answer, or give an example.

Question 4: Do you think it is possible to come up with 3 points such that it's impossible to draw a parabola through them?  Explain your answer, or give an example.

Part II: How Does the Vertex Location of a Parabola Change?

Open up the How Does the Vertex Location of a Parabola Change? demonstration.

Again you have a parabola and sliders for a, b, and c, only this time they are fixed in the coordinate plane.  Check "Label" to see the equation of the parabola.   If you check "Show", a mirror-image parabola shows up in red.  Play with the 3 sliders for a while, observing how the the vertex location changes.

Question 5: Describe how changing each coefficient changes the vertex of the parabola.  Be as specific as possible!
(a)
(b)
(c)

Part III: Sliding the Roots of Quadratics

Open up the Sliding the Roots of Quadratics demonstration.

What you're looking at is the complex plane.  Remember, the real axis is horizontal and the imaginary axis is vertical.  Thus, numbers on the real axis are real; all other numbers are imaginary, like 2 + 3i.  (Numbers on the imaginary axis are "pure imaginary," like 2i.)

There is also an equation on top.  The 2 dots are solutions to that equation.

Slide the sliders around.

Question 6: Which slider controls b?  Which one controls c?

There are 3 possibilities for the solutions: 2 reals, 2 complex #s that are conjugates of each other, or one real #.

Slide the sliders around some more, and make the following table, including at least 8 rows, and at least one of EACH of the 3 possibilities listed above.

b	c	b^2 - 4ac	solutions (2 real, 2 imaginary, or 1 real)

Question 7: When does the quadratic equation have only one solution?  (Hint: Think about the discriminant, b^2 - 4ac.)  When does it have 2 real solutions?  When does it have 2 imaginary solutions?  (You might have to add more rows to your table to figure it out.)

Question 8: Explain why, when looking at the solutions or zeros of a quadratic function, we don't need to have a slider for a.

Part IV: The Multiplication Parabola

Open up the Multiplication Parabola demonstration.

The parabola you're looking at is y = x^2.

The 2 points plotted on the parabola are (a, a^2) and (b, b^2), where a is negative and b is positive. 

Question 9: Explain how you know for sure that these 2 points are on the graph of y = x^2.

There is a line segment connecting the 2 points, and the midpoint, they say, is (0, -ab).  In other words, you can do the multiplication graphically!

Question 10: Use the parabola to compute (.8)(1.2).  Did it work?

Now we are going to prove that this works!  For this you don't need a computer, but you might want to work with a partner.  You may do question 11 on paper, instead.

Question 11: Proof
Notice that the point in the middle, (0, -ab), is the y-intercept of the line through the 2 points.  We're going to call it (0, k) and prove that k = -ab.
(a) Calculate the slope of the line through (a, a^2) and (b, b^2).  Remember to factor the numerator in order to reduce the fraction!
(b) Use the slope intercept form of the equation y = mx + k, along with the slope you just calculated and one of the points, to solve for k.
Ta-da!  You should have gotten k = -ab.

Conclusion
When you get done, poke around the Demonstrations Page to find some interesting math demonstrations.  You can use the search function to find demos on specific topics, or you can just browse.