Now, let's look at some descriptions and how they relate to other representations.

• A quadratic function has roots at -2 and 3 with a maximum point.

The roots indicate the x-intercepts (zeros) of the graph of the function.

The maximum point is the vertex and indicates the parabola opens downward, since for the parabola to open upward, the vertex has to be the lowest point (minimum) on the graph.

A sketch of the graph might look like this:

By comparing the points on the graph to the values in the table, you can see the coordinates of the x-intercepts (zeros) are (-2, 0) and (3,0). The table shows the y-values increasing from -3 to 3, then repeating the other direction, showing the vertex occurring between x values of 0 and 1, or at 0.5, and the y-coordinate just above 3, since the two values on either side of the maximum are both exactly 3.

Since the zeros occur when x = -2 and x = 3, the equation can be written in factored form using the zeros to write the factors of the quadratic. In order to have a parabola that has a maximum (opens downward), the coefficient of the x² term must be negative.

Is there only one possible quadratic function that models the description given above? Study the following graph.

Which of these graphs model the description?

Interactive popup. Assistance may be required.

Check Your Answer

They all do. They all have x-intercepts at -2 and 3 and have a maximum point.

These graphs differ in the value of the coefficient of x² or “a”

• Graph A is the parent function with the a value -1: y = (x + 2)(x - 3)
• Graph B is stretched in the y direction by an a value of -2: y = -2(x + 2)(x - 3)
• Graph C is compressed in the y direction by an a value of -1 over 2 1 2 : y = -1 over 2 1 2 (x + 2)(x - 3)

Are there others? Try entering the three equations into your graphing calculator