Quadratic function rules can be expressed in factored form, sometimes referred to as intercept form.
As is the case with standard form and vertex form, gives the direction of the parabola. When the parabola faces upward, and when it faces downward. Additionally, the zeros of the parabola lie at Because the points of a parabola with the same -coordinate are equidistant from the axis of symmetry, the axis of symmetry lies halfway between the zeros. Consider the graph of
From the graph, the following characteristics can be connected to the function rule.
When quadratic functions are written in factored form some of the parabola's characteristics are easy to identify. It's possible to graph a quadratic function using its characteristics. Consider the function
To begin, identify the zeros, and from the function rule. Since the rule is Thus, the zeros are and Next, plot these points on a coordinate plane.
Points on the parabola with the same -coordinate are equidistant from the axis of symmetry. That means, the axis of symmetry is located halfway between the zeros. Notice that the zeros lie units apart.
Thus, the axis of symmetry is units away from both of the zeros. Moving units from either zero toward the other yields This can be verified algebraically using Draw the axis of symmetry at
Considering the direction of the parabola, given by the -value in the function rule, the shape of the graph can be seen. Here, Thus, the parabola opens downward. To draw the parabola, connect the points with a smooth curve.
Allister and Lebowski spend a Sunday afternoon launching bottle rockets in their backyard. For one round of launches, they collect the following data. The trajectory of Allister's rocket can be modeled by the function while Lewbowski's is modeled by Note that and are both given in meters. The graph of is shown.
Determine whose rocket went higher and whose went further.
Here, we will compare two quadratic functions that model the trajectory of each rocket. To find the maximum height we'll need the vertex, and to find the horizontal distance we'll need the zeros. We'll begin with Lebowski's rocket.