We explore quadratic functions.

Quadratic Graphs

The function in the above story problem is an example is a quadratic function.

The graph of a quadratic function is called a parabola. Notice the symmetry in the table, how the -values in rows and match as well as rows and ? Also notice the symmetry in the shape of the graph, how its left side is a mirror image of its right side. The parabola opens upward or downward according to the sign of the leading coefficient . If the leading coefficient is positive, the parabola opens upward. If the leading coefficient is negative, the parabola opens downward.

You can play with changing the value of on the graph using Desmos and see how that changes the parabola.

The vertex of a parabola is the highest or lowest point on the graph, depending uponon whether the graph opens downward or upward. In Example 1, the vertex is . This tells us that Hannah’s rocket reached its maximum height of feet after seconds. If the parabola opens downward, as in the rocket example, then the -value of the vertex is the maximum -value. If the parabola opens upward then the -value of the vertex is the minimum -value. The axis of symmetry is a vertical line that passes through the vertex, cutting the parabola into two symmetric halves. We write the axis of symmetry as an equation of a vertical line so it always starts with “”. In Example 1, the equation for the axis of symmetry is .

It is possible for a quadratic graph to have zero, one, or two -intercepts.

In Example 1, first note that this is a function . We will have -intercepts but rather than having -intercepts, these will be -intercepts due to our use of the variable and not . The point is the starting point of the rocket, and it is where the graph crosses the -axis, so it is the -intercept. The -value of means the rocket was on the ground when the -value was , which was when the rocket launched. The point on the path of the rocket is also a -intercept. The -value of indicates the time when the rocket was launched from the ground. There is another -intercept at the point , which means the rocket came back to hit the ground after seconds.