You have studied the standard form and factored form of the equation of a parabola. Answer the following questions about parabolas. Click “Submit” when finished.

You have seen how to write the equation of a parabola in standard form and in factored form. Now you will learn another form for the equation of a parabola. Look at the table and graph displayed. The vertex of this parabola is <EQUATION>. This point is also the parabola's only <EQUATION>intercept, so the equation of the parabola has <EQUATION> as a factor twice. So the equation of this parabola is <EQUATION>.

Now consider the equation <EQUATION>. Notice that this equation is not in factored form because it has an extra term, “<EQUATION>”. The vertex of this parabola is <EQUATION>. This parabola has no <EQUATION>, so its equation cannot be written in factored form. Do you notice a relationship between the equation and the coordinates of the vertex? Look at the graphs of the two parabolas. Notice that the vertex of <EQUATION> is <EQUATION> units above the vertex of <EQUATION>. Both of these equations are written in |B| vertex form |/B| . In general, this form of a quadratic equation is written <EQUATION>.

Let's explore more quadratic equations of the form <EQUATION>. Use the sliders to change the values of <EQUATION> in the equation graphed. Notice how these values affect the location of the axis of symmetry, the location of the vertex, the number of <EQUATION>, and the location of the <EQUATION>. Click “Done Exploring” when finished.

When you have a quadratic equation in the form <EQUATION>, the vertex of the parabola is <EQUATION>. The values of <EQUATION> also affect the number and the location of the <EQUATION>. The value of <EQUATION> determines the location of the axis of symmetry because the value of <EQUATION> moves the parabola only horizontally. The value of <EQUATION> moves the parabola only vertically.

The graph of the equation <EQUATION> is displayed. Let's write this equation in vertex form. The coefficient of <EQUATION>, so we know that <EQUATION>. The vertex of the parabola is <EQUATION>. We can substitute the <EQUATION> - and <EQUATION> of the vertex for <EQUATION> to find the vertex form. The vertex form is <EQUATION>. We can also write an equation for this parabola in factored form. Again we replace a with <EQUATION>. We can then use the <EQUATION> to find the factors.

Look at the graph of the parabola. Use the vertex and the <EQUATION> to select any of the equations that could describe the parabola. Click “Submit” when finished.

Look at the graph of the parabola. Use the axis of symmetry and the intercept at <EQUATION> to select any of the equations that could describe the parabola. Click “Submit” when finished.

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