Changing Values in Parabolas

By Amber Candela

Assignment 3

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The problem:

Using the above parabola, fix two values for a, b and c. Make multiple graphs by varying c. Then fix two other values and vary the third. What observations are noticed?

Let's vary "c"!

First I varied the value for "c"

When "c" is varied, the graph is translated up or down. When c = -5 the -intercept = -5. The graph then is translated two units up to where the -intercept = -3 and the graph continues to be translated two units up each time until c = 3 where the -intercept =3. Starting at c = -5 and ending at c = 3, the graph is translated up a total of 8 units.

Now on to "b"

I then varied the values of "b"

The "b" value determines where the vertex and axis of symmetry of the graph will be. The equation for the axis of symmetry is x = -b/2a. Since all the values of "a" are equal to one in this case the axis of symmetry is x = -b/2. The first graph starts with b = 1 initial axis of symmetry is x = -1/2. As the"b" changes the axis of symmetry changes from x = -1/2 to x = 1 to x = -3/2 to x = 2 to x = -5/2 to x = 3 to x = 3 to x = -7/2 to x = 4. If "b" is negative the vertex will be positive, if "b" is positive the vertex will be negative. The -intercept for each graph is equal to one.

Moving on to "a"

I then varied the values of "a".

If "a" > 1, the larger "a" becomes, the width of the parabola becomes smaller. If 0 < a < 1, the closer to zero "a" gets the wider the parabola becomes. If "a" is positive the parabola opens upward, if "a" is negative the parabola opens downward. The y-intercept = -3 for each parabola.

What if a = 0?.

When a = 0, you get the linear equation y = 2x - 3. The y intercept is -3 which happens to be the point where all the parabolas cross the y - axis.