High School Geometry Unlocked (2016)

Lesson 6.3. Parabolas

A parabola is a U-shaped curve. It belongs to a group of objects called conic sections—literally, cross-sections of cones. A parabola is also what you get when you graph a quadratic equation—an equation in which one of the terms is raised to a power of 2 (for example, y = x² + 2x + 4).

The vertex is the point at which the parabola changes direction—that is, the curve swings from down to up, or left to right. All parabolas have a line of symmetry—called the axis of symmetry—that passes through the vertex.

The definition of a parabola involves a point, called the focus, and a line, called the directrix. The parabola is the set of points that are equidistant from the focus and the directrix. In other words, if you have a point on a parabola (call the point X), the straight-line distance between X and the focus is equal to the perpendicular distance between X and the directrix.

Note that the axis of symmetry of a parabola is perpendicular to the directrix, and passes through the focus and vertex. The vertex is always halfway between the focus and directrix.

GRAPHING A PARABOLA FROM AN EQUATION

Equation of a Parabola: Standard Form

You may have had some experience graphing parabolas in algebra class. One standard method is to first factor the quadratic equation, and solve for the roots (also known as zeroes) of the equation—that is, the x-values that make y = 0. This gives two points on the parabola, which you can then use to find the vertex. For the purposes of this lesson, we’ll assume that you’re somewhat familiar with factoring quadratic expressions. If not, you may find it helpful to brush up a little before proceeding.

Graph the parabola represented by the equation y = x² + 8x + 15.

Try factoring the quadratic expression. Begin by finding the factors of 15:

1, 15

3, 5

Now, find the pair of factors that adds up to 8.

That means that the binomial factors of x² + 8x + 15 are (x + 3) and (x + 5).

y = x² + 8x + 15

y = (x + 3)(x + 5)

Now, find the values for x where y = 0.

0 = (x + 3)(x + 5)

That means that either (x + 3) or (x + 5) equals zero.

If (x + 3) = 0, then x = −3.

If (x + 5) = 0, then x = −5.

Plugging in either of those two values into the original equation will make y = 0.

y = (x + 3)(x + 5)

= (−3 + 3)(−3 + 5)

= (0)(2)

= 0

and

y = (x + 3)(x + 5)

= (−5 + 3)(−5 + 5)

= (−2)(0)

= 0

Therefore, the two roots of the equation are x = −3 and x = −5. In other words, the coordinates (−3, 0) and (−5, 0) exist on the parabola, where the parabola intersects the x-axis.

The parabola’s vertex will have an x-value that is exactly halfway between the two zeroes. (This is also true using any two points that have the same y-value on the parabola). The coordinate that’s exactly between (−3, 0) and (−5, 0) is (−4, 0).

Therefore, the x-coordinate of the vertex is −4. To find the corresponding y-value, plug −4 into the equation.

y = (x + 3)(x + 5)

y = (−4 + 3)(−4 + 5)

y = (−1)(1)

y = −1

Therefore, when x = −4, y = −1. This gives us the coordinate (−4, −1), which is the parabola’s vertex. With the vertex and two zeroes, we can graph the parabola:

You may feel more comfortable with your graph if you plug in to find a few more points. For example, if x = −1, then y = 8 ((−1 + 3)(−1 + 5)); and if x = −7, then y = 8 ((−7 + 3)(−7 + 5)).

Here is how you may see parabolas on the SAT.

What is the vertex of the parabola defined by the equation y = x² + 4x − 12?

A.(0, −12)

B.(−6, 0)

C.(−2, −16)

D.(2, −12)

Equation of a Parabola: Vertex Form

For a parabola with vertex (h, k):

An equation in vertex form provides specific information—namely, the coordinate of the vertex. If you see an equation in this form, you can know immediately that the vertex is (h, k).

However, the vertex by itself is not enough to tell us the shape of the parabola. To plot the parabola, you’ll need at least three points. Therefore, choose a couple of different values for x and find the corresponding y-coordinates.

Graph the parabola with the equation y = 2(x − 4)² + 3.

Since this equation is in vertex form, you know that the vertex of the parabola is (4, 3).

To graph the rest of the parabola, you’ll need a couple more points. To find additional points that lie on the parabola, you can choose different values for x and then evaluate for the corresponding y-coordinates.

Let’s try x = 2.

Therefore, the point (2, 11) lies on the parabola.

Try one additional value for x. We’ll use x = 6.

Therefore, the point (6, 11) also lies on the parabola.

With three points, the parabola can be graphed:

You can find the vertex by remembering that a parabola is symmetrical. If you find the x-values for a given y-value, the midpoint between the x-values will be the same as the x-value at the vertex. The easiest point to find is when y = 0. You can factorx² + 4x − 12 = 0 as (x + 6)(x + 2) = 0. Therefore, x = −6 or 2; the midpoint is the average of the x-values: = − 2. Only (C) has an x-coordinate of −2. The correct answer is (C).

You can also derive the focus and directrix from the vertex form of an equation.

Focus and Directrix from Vertex Form

For a parabola with vertex (h, k):

The value is equal to the distance between the vertex and the focus. This is also equal to the distance between the vertex and the directrix.

Therefore, if you have a parabola in vertex form, you can add the quantity to the vertex coordinate in order to find the focus.

Likewise, you can subtract from the vertex coordinate to find the y-coordinate of the directrix (for a vertical parabola).

Find the vertex and directrix for the parabola with the equation y = (x − 4)² + 3.

This equation is already in vertex form, and you know that a = . Find the quantity :

Therefore, the distance between the vertex and the focus is 2. The vertex is (4, 3), from the equation. Since this is a vertical parabola (it’s in the form of y =), just add 2 to the y-coordinate.

Vertex = (4, 3)

Focus = (4, 5)

The directrix, then, will be two units below the vertex, so it passes through the coordinate (4, 1). Remember that the directrix is a line, which in this case is the line y = 1.

Directrix: y = 1

FINDING THE EQUATION OF A PARABOLA

Find the Equation, Given the Zeroes

To find the equation of a parabola from its graph, try to start with the zeroes—that is, the points where the parabola crosses the x-axis. This will give you two factors of the equation. For the purposes of this exercise, let’s say that we have two points on the parabola at (m, 0) and (n, 0). Then, we know that two factors are (x − m) and (x − n).

However, that’s not enough to derive the full equation, because there are actually an infinite number of parabolas that can pass through those two points.

Therefore, we need a third point in order to find the missing factor, a. You’re going to put your equation in the following form, and plug in the values of a known coordinate to solve for a.

y = a (x − m)(x − n)

Find the equation of the parabola with zeroes at (−5, 0) and (3, 0), and the vertex (−1, −4).

If the zeroes are (−5, 0) and (3, 0), then you know that two factors of the equation are (x + 5) and (x − 3).

y = a (x + 5)(x − 3)

To solve for a, plug in the coordinates of the vertex (that is, x = −1 and y = −4).

Therefore, the equation of the parabola is y = (x + 5)(x − 3).

Find the Equation, Given the Focus and Directrix

Recall that any point on a parabola is equidistant from the focus and the directrix. That is, if you have a point (x, y), then the distance to the focus is equal to the distance to the directrix. Therefore, to derive the equation of a parabola, we can just find those two distances and set them equal to each other.

In the figure above, the focus of the parabola is (a, b) and the directrix is the line y = c. Take a point on the parabola, (x, y). To calculate the distance from (x, y) to (a, b), use the distance formula or Pythagorean theorem:

Distance from (x, y) to (a, b):

d =

Now, find the distance between (x, y) and the directrix. Recall that the shortest distance between a point and a line will be through a segment perpendicular to the line, as shown:

Which means that the distance between (x, y) and the directrix is |y − c|. (Use absolute value, because distances are always positive).

Now, set the values equal to each other, and simplify. Yes, this will have a lot of steps! Work carefully, and don’t do anything in your head.

That’s quite a bit of algebra, there, and it can still be simplified further! But this format is fine, and it has y by itself.

Equation of a Parabola, with Focus and Directrix

Given focus (a, b) and directrix y = c: