Conics
Parabolas
Learning Objectives
By the end of this section, you will be able to:
- Graph vertical parabolas
- Graph horizontal parabolas
- Solve applications with parabolas
Before you get started, take this readiness quiz.
Graph Vertical Parabolas
The next conic section we will look at is a parabola. We define a parabola as all points in a plane that are the same distance from a fixed point and a fixed line. The fixed point is called the focus, and the fixed line is called the directrix of the parabola.
A parabola is all points in a plane that are the same distance from a fixed point and a fixed line. The fixed point is called the focus, and the fixed line is called the directrix of the parabola.
Previously, we learned to graph vertical parabolas from the general form or the standard form using properties. Those methods will also work here. We will summarize the properties here.
| Vertical Parabolas | ||
|---|---|---|
| General form |
Standard form |
|
| Orientation | up; down | up; down |
| Axis of symmetry | ||
| Vertex | Substitute and solve for y. |
|
| y-intercept | Let | Let |
| x-intercepts | Let | Let |
The graphs show what the parabolas look like when they open up or down. Their position in relation to the x– or y-axis is merely an example.
To graph a parabola from these forms, we used the following steps.
- Determine whether the parabola opens upward or downward.
- Find the axis of symmetry.
- Find the vertex.
- Find the y-intercept. Find the point symmetric to the y-intercept across the axis of symmetry.
- Find the x-intercepts.
- Graph the parabola.
The next example reviews the method of graphing a parabola from the general form of its equation.
Graph by using properties.
| Since a is the parabola opens downward. | |
| To find the axis of symmetry, find | |
| The axis of symmetry is | |
| The vertex is on the line | |
| Let | |
| The vertex is | |
| The y-intercept occurs when | |
| Substitute | |
| Simplify. | |
| The y-intercept is | |
| The point is three units to the left of the line of symmetry. The point three units to the right of the line of symmetry is |
Point symmetric to the y-intercept is |
| The x-intercept occurs when | |
| Let | |
| Factor the GCF. | |
| Factor the trinomial. | |
| Solve for x. | |
| The x-intercepts are | |
| Graph the parabola. |
Graph by using properties.
Graph by using properties.
The next example reviews the method of graphing a parabola from the standard form of its equation,
Write in standard form and then use properties of standard form to graph the equation.
| Rewrite the function in form by completing the square. |
|
| Identify the constants a, h, k. | , , |
| Since the parabola opens upward. | |
| The axis of symmetry is | The axis of symmetry is |
| The vertex is | The vertex is |
| Find the y-intercept by substituting | |
| y-intercept | |
| Find the point symmetric to across the axis of symmetry. | |
| Find the x-intercepts. | |
| The square root of a negative number tells us the solu