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 Write-up

Properties of Parabolas
Problem Statement
Compare the graphs of quadratic functions in the form: y = ax2 + bx + c.

a. What are the coordinates of the y-intercept?
b. What must be true in order for the graph to reflect over the y-axis?
c. What must be true in order for the graph to open down? to open up?

 Extensions

 d. What must be true in order for the graph to touch the x-axis only one time? Zero times? e. How can you calculate the equation of the axis of symmetry (the vertical line where the graph can be folded in half)? f. Where on the graph is the solution for ax2 + bx + c = 0 ?

 Related External Resources

 Exploring Parabolas - This java demonstration shows a geometric construction of a parabola. [ java applet ]

Problem setup

This exploration wants me to determine what happens to the graph when certain parameters of a Quadratic equation changes.

Plans to Solve/Investigate the Problem

I used my handy dandy TI 84 Silver Edition (can you tell how proud I am of the calculator?) to determine what would happen.  I entered the following Quadratic Equations

Below, you will find a graph of the functions.  You will have to imagine that the Y axis appears on the gridline that is marked 0 along the horizontal axis.

Notice that when my constant term changes, my graph is shifted up the number of positive units and down for the number of negative units.

Part B What must be true in order for the graph to reflect over the y-axis?

Here, if you look at the equations used, the only difference is in the sign of the second term.  Now, IMAGINE that the Y axis is at the 0 gridline.  You will hopefully observe that the curves are a reflection of each other.  So it appears that changing the sign of the middle term will reflect your parabola across the Y Axis!  HOW EXCITING!!

Part C What must be true in order for the graph to open down? to open up?

Take a look at the Equation Box.  How do the equations compare? The only difference in the two is the sign on the LEADING COEFFICIENT!

So, it appears that the sign of the leading coefficient determines the direction of the parabola!

Part D. What must be true in order for the graph to touch the x-axis only one time? Zero times?

For this, I will refer to my pre existing knowledge and you can take a look at the link, to get a quick lesson on FACTORING.

When you factor a QUADRATIC EQUATION, you can go one step further to determine the

X Intercepts. (X intercepts are the places where the graph crosses the X Axis)

Look at the following factoring thingy! (Give me a break!  You KNOW what a THINGY is!!)

The only time that the Parabola will just touch the axis once is when you are dealing with a PERFECT SQUARE TRINOMIAL (PST).  In the Factoring Thingy, you should observe that the factored form has 2 identical factors.  Recall that when you multiply a thing by itself, you are really SQUARING that thing.  Since (x+9) (x+9) = (x+9)2, the original equation is considered a PST.  The root (another term for the x intercepts) is actually a double root since it appears twice when solving.  Hence, there is only one intercept!

You will have to forgive my laziness.  I just really don’t feel like explaining the line of symmetry thing.  Take a look at this link, and amuse yourself! The equation that goes through the vertex, is the line of symmetry.

Part F Where on the graph is the solution for ax2 + bx + c = 0 ?

If you read carefully, you will see that I have already answered this question!  Factoring and using the ZERO PRODUCT PROPERTY yields the Zeros – Roots – X intercepts of a Quadratic.  If a Quadratic is not factorable, utilize the QUADRATIC Formula.

Investigation/Exploration of the Problem

Author & Contact
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Link(s) to resources, references, lesson plans, and/or other materials
PurpleMath.com