Assignment 2

 

By:  J. Matt Tumlin, Cara Haskins, & Robin Kirkham

 

Examine graphs for the parabola y = ax2 + bx + c for different values of a, b, and c (a, b, c can be any rational numbers).

 

Our first step will be to look at the basic parabola

when a=1, b=0 and c=0.

 

 

 

Notice that the Domain is the set of real numbers, and the Range for this basic parabola is all non-negative numbers.

 

The lowest point on a parabola is called the minimum.  The minimum point on our basic parabola is (0,0).

 

Continue to use the basic quadratic function as our frame of reference. Let us examine what happens to the graph under the following guidelines.

 

Step 1:  Let b=0,c=0, and vary the values of a. Our new equation becomes

y=ax2 .

 

Let us use Graphing Calculator 3.2 to examine the effects of using different values for a, remembering to use positive and negative values.

 

 

 

The red graph is y= x2. This basic parabola will always be in red in future examples for comparison purposes.

 

Notice that the minimum of the graphs does not change even though the value for a was varied.  When a > 1, the graph has been narrowed horizontally, resulting in a horizontal shrinking of the graph.  When 0 < a < 1, the graph has now been stretched horizontally.

 

This leaves the question what happens when negative values are substituted for variable a?

 

 

 

By substituting negative values for a, notice there is a reflection across the x-axis for our two graphs as well as a horizontal change of the basic parabola.

 

The highest point on a parabola is called the maximum. The maximum point on our two parabolas is (0,0).

 

Step 2: Now we are examining the effects of variable b.  Let a=1, and c=0 and change the values for b.  Our new equation is now:

y = x2 + bx

 

 

 

Notice that the widths of the parabolas stayed the same, while the location of the minimum changed.  This movement appears to be equal to the value of -b/2, both vertically and horizontally.  This is investigated in our Step 3 below.

 

Step 3: Let us start again with our original equation y= ax2 + bx + c. Let a=1, b=0, and vary c, resulting in:

y = x2 + c

 

 

 

The value of variable c moves the parabola up or down. When c > 0, the graph moves to up.  When c < 0, the graph moves down.

 

This vertical movement changes in respect to our minimum point.  The vertical shift appears to be equal to the value of c.

 

To be sure, let us check what happens to a change of variable b and c simultaneously.

 

 

 

This shows us the horizontal and vertical shifts are a result of both the variables b and c respectively.  The horizontal shift still appears to be -c/2, while the vertical shift appears to be smaller than c.

 

To be sure, let us investigate what happens to a change of variable a ,b and c simultaneously.

 

 

 

This shows us the horizontal and vertical shifts are a result of all the variables a, b and c respectively.  The horizontal shift turns out to be b/2a, while the vertical shift turns out to be the y value when x is equal to b/2a.

 

 This set of parabolas introduces many introduces many interesting effects.  Firstly, one can see y= ax2 + bx + c where a, b, and c are all positive and the similar parabola where a is the additive inverse.  One observes that these two parabolas are inverses and both shift to opposite quadrants.

 

In summary, given the equation y = ax2 + bx + c the following are true:

 

      Changes in the value of a effects the direction and width of the parabola.

 

      Changes in the value of b effects the horizontal and vertical shift.

 

      Changes in the value of c effects the vertical shift.

 

 

      Changes in the value of a, b and c together effect the total shift of the parabola.

 

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