Graphing Parabolas with Vertices at the Origin
In The Ellipse, we saw that an ellipse is formed when a plane cuts through a right circular cone. If the plane is parallel to the edge of the cone, an unbounded curve is formed. This curve is a parabola.
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Set the two expressions for equal to each other and solve for to derive the equation of the parabola. We do this because the distance from to equals the distance from to .
We then square both sides of the equation, expand the squared terms, and simplify by combining like terms.
The equations of parabolas with vertex are when the x-axis is the axis of symmetry and when the y-axis is the axis of symmetry. These standard forms are given below, along with their general graphs and key features.
A General Note: Standard Forms of Parabolas with Vertex (0, 0)
The table below and Figure 5 summarize the standard features of parabolas with a vertex at the origin.Axis of Symmetry | Equation | Focus | Directrix | Endpoints of Latus Rectum |
x-axis | ||||
y-axis |
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How To: Given a standard form equation for a parabola centered at (0, 0), sketch the graph.
- Determine which of the standard forms applies to the given equation: or .
- Use the standard form identified in Step 1 to determine the axis of symmetry, focus, equation of the directrix, and endpoints of the latus rectum.
- If the equation is in the form , then
- the axis of symmetry is the x-axis,
- set equal to the coefficient of x in the given equation to solve for . If , the parabola opens right. If , the parabola opens left.
- use to find the coordinates of the focus,
- use to find the equation of the directrix,
- use to find the endpoints of the latus rectum, . Alternately, substitute into the original equation.
- If the equation is in the form , then
- the axis of symmetry is the y-axis,
- set equal to the coefficient of y in the given equation to solve for . If , the parabola opens up. If , the parabola opens down.
- use to find the coordinates of the focus,
- use to find equation of the directrix,
- use to find the endpoints of the latus rectum,
- If the equation is in the form , then
- Plot the focus, directrix, and latus rectum, and draw a smooth curve to form the parabola.
Example 1: Graphing a Parabola with Vertex (0, 0) and the x-axis as the Axis of Symmetry
Graph . Identify and label the focus, directrix, and endpoints of the latus rectum.Solution
The standard form that applies to the given equation is . Thus, the axis of symmetry is the x-axis. It follows that:- , so . Since , the parabola opens right the coordinates of the focus are
- the equation of the directrix is
- the endpoints of the latus rectum have the same x-coordinate at the focus. To find the endpoints, substitute into the original equation:
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Try It 1
Graph . Identify and label the focus, directrix, and endpoints of the latus rectum. SolutionExample 2: Graphing a Parabola with Vertex (0, 0) and the y-axis as the Axis of Symmetry
Graph . Identify and label the focus, directrix, and endpoints of the latus rectum.Solution
The standard form that applies to the given equation is . Thus, the axis of symmetry is the y-axis. It follows that:- , so . Since , the parabola opens down.
- the coordinates of the focus are
- the equation of the directrix is
- the endpoints of the latus rectum can be found by substituting into the original equation,
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Try It 2
Graph . Identify and label the focus, directrix, and endpoints of the latus rectum. SolutionWriting Equations of Parabolas in Standard Form
In the previous examples, we used the standard form equation of a parabola to calculate the locations of its key features. We can also use the calculations in reverse to write an equation for a parabola when given its key features.How To: Given its focus and directrix, write the equation for a parabola in standard form.
- Determine whether the axis of symmetry is the x- or y-axis.
- If the given coordinates of the focus have the form , then the axis of symmetry is the x-axis. Use the standard form .
- If the given coordinates of the focus have the form , then the axis of symmetry is the y-axis. Use the standard form .
- Multiply .
- Substitute the value from Step 2 into the equation determined in Step 1.
Example 3: Writing the Equation of a Parabola in Standard Form Given its Focus and Directrix
What is the equation for the parabola with focus and directrixSolution
The focus has the form , so the equation will have the form .Multiplying , we have . Substituting for , we have .