You have studied how to factor special products, including a difference of squares. Answer the following questions about differences of squares. Click “Submit” when finished.

We want to see the shape of the graph of the equation <EQUATION> Let's pick some <EQUATION> values and find the corresponding <EQUATION> values.

Now we plot each pair of <EQUATION> and <EQUATION>values. The set of all points that satisfy the equation <EQUATION> is a curve called a |B| parabola |/B| .

Let's find the slope between some of the points on the parabola. The slope between the first pair of points is <EQUATION>, or <EQUATION>. The slope between the second pair of points is <EQUATION>, or <EQUATION>. The slope between these two pairs of points is not the same. So, the slope between points on a parabola is not always the same. We can see from the graph that the slope between some points will be negative, and the slope between other points will be positive. This could never occur between points on a line. Remember that on a line, the slope between any pair of points is always the same.

From the graph of the parabola, we can see that the parabola has a minimum value. This point on the parabola is called the vertex. The |B| vertex |/B| of this parabola is (0, 0). The points <EQUATION> and <EQUATION> are both at the same height and they are each <EQUATION> units from the <EQUATION>axis. The points <EQUATION> and <EQUATION> are also at the same height, and are each <EQUATION> units from the <EQUATION>axis. And, the points <EQUATION> and <EQUATION> are at the same height, and they are each <EQUATION> unit from the <EQUATION>axis. Every point to the left of the <EQUATION>axis has a corresponding point to the right of the <EQUATION>axis. When the left half of a figure matches the right half, the figure is called |B| symmetric |/B| . The line that separates the two halves is called the |B| axis of symmetry |/B| . In this case, the <EQUATION>axis is the axis of symmetry. The vertex will always lie on the axis of symmetry.

Here is another parabola. It crosses the <EQUATION> twice, so it has two <EQUATION> The parabola has one <EQUATION> Notice that in this case the vertex is the maximum value of the parabola, rather than the minimum value. This is true for all downward-facing parabolas.

Here is another parabola. This parabola touches the <EQUATION> only once. It has one <EQUATION> and one <EQUATION>

Here is a parabola that does not cross the <EQUATION> axis at all. The vertex has the minimum <EQUATION>value, and the vertex is above the <EQUATION> axis. So, the parabola doesn't have any <EQUATION> intercepts. It does have one <EQUATION> intercept.

Here are the four parabolas you have seen so far. Although these parabolas are located at different places in the coordinate plane, they all have the same general shape. You can see that each parabola has a vertex and an axis of symmetry. And each parabola has one <EQUATION> intercept. The number of <EQUATION> intercepts is not the same in all parabolas. Parabolas can have <EQUATION>, <EQUATION> or <EQUATION> <EQUATION> intercepts.

Here is another parabola. We can see that this parabola has one <EQUATION> at <EQUATION> and another at <EQUATION>. The axis of symmetry is <EQUATION>. The parabola is symmetric about this vertical line. So, every point to the left of this line has a matching point to the right of the line. One the <EQUATION> is one unit to the left of the axis of symmetry, and the other <EQUATION> is one unit to the right of the axis of symmetry.

Here is a parabola with two <EQUATION> intercepts. The <EQUATION> intercept on the left is at <EQUATION>. The axis of symmetry of this parabola is <EQUATION>. Enter the coordinates of the second <EQUATION> intercept. Click “Submit” when finished.

Copyright 2006 The Regents of the University of California and Monterey Institute for Technology and Education