Tiffany N. KeysŐ Assignment 2:

Look at
that Parabola!

A quadratic function has the form y = *a*x^{2}
+ *b*x +*c*, where a is not equal to zero. The U-shaped graph of a
quadratic function is called a parabola. The
graphs of all quadratic functions are related to the graphs y = x^{2 }and
y = - x^{2}

y = x^{2 }y = - x^{2}

These
graphs have the following characteristics:

¤
The
**origin** is the lowest point on the
graph of y = x^{2} and the highest point of the graph of y = -x^{2}

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The
lowest point or the highest point on the graph of a quadratic function is
called the **vertex**.

¤
The
**axis of symmetry** for the graph of a
quadratic function is the vertical line through the vertex. The graphs above
are both symmetric about the axis of symmetry or, in these cases, the y –
axis.

*INVESTIGATION*: Construct graphs for the parabola y = *a*x^{2}
+ *b*x +*c* for different
values of a, b, and c.

First,
letŐs begin with substituting in values for *a*:

**Y = 3x ^{2} **

**Y =
-3x ^{2}**

OBSERVATIONS:

á
The
parabola always passes through the origin.

á
The
coordinates of the vertex of the parabola do not change in each equation as
different values for a are substituted in.

á
The
concavity of the parabola is the aspect of the graph changes in each equation.
As a increases, the concavity of the parabola
decreases. If a is negative, then it is
reflected across the x-axis.

Next,
letŐs observe what happens when different values for *b*
are substituted in:

**Y = x ^{2}+3x**

**Y = x ^{2}-
3x **

OBSERVATIONS:

á
As
above, the parabola always passes through the origin.

á
However,
the coordinates of the vertex of the parabola are the aspect of the graph that
changes in each equation. As b increases, the
vertex shifts across the x-axis. If b is
negative, then it reflected across the y-axis.

á
The
concavity of each parabola does not change in each equation as different values
of b are substituted in.

Lastly,
letŐs notice the difference in the graph when values for *c* are substituted in:

**Y = x ^{2}+ x + 3**

**Y = x ^{2}+ x - 3 **

OBSERVATIONS:

á
The
parabola does not pass through the same point as c
changes.

- The coordinates
of the vertex of the parabola are the aspect of the graph that changes in
each equation. As c increases, the
vertex, depending on if c is positive or
negative, moves up or down along the y-axis.
- The concavity of
the parabola is another aspect of the graph changes in each equation. As c increases, the concavity of the parabola
increases.