# Parabolas

## Parabolas

A **parabola** is the distinct shape of a **quadratic equation** when it is graphed. In this video, we’ll give you some clues found in quadratic equations that can tell you more about the parabolas they graph.

So, the symmetry of a parabola is due to the squared term in the quadratic equation. However, there is one point on the parabola that does not have a “mirror image.” This point is called the **vertex**, and is notated by the ordered pair, (h,k). The vertex is either the **maximum or minimum** value of the parabola, depending on which direction the parabola opens.

When the **x-variable is squared** in a quadratic equation, the parabola will “open” either **up or down**. Picture the two symmetric branches of the graph extending away from the vertex in the same direction; either up, towards positive infinity on the y-axis, or down, towards negative infinity on the y-axis.

However, when the **y-variable** is squared in a quadratic equation, the parabola will “open” either to the **right**, as both branches extend away from the vertex towards positive infinity on the x-axis, or to the **left**, towards negative infinity on the x-axis.

To understand this further, we need to look at how quadratic equations are written. They are written using two basic forms:

- 1. A

**standard form**quadratic equation is written as, y=ax

^{2}+bx+c, where “a” represents the coefficient of the squared variable, “b” is the coefficient of the linear term, and “c” represents a constant. Because the x-variable is squared, this is a quadratic function which opens either up or down.

A quadratic equation in this form with the y-variable squared would be written as, x=ay^{2}+by+c. The parabola graphed from this relation would open either left or right.

- 2. A quadratic that is in

**vertex form**is written as, y=a(x-h)

^{2}+k, where “a” is the coefficient of a squared binomial which, in this example, contains the x-variable. The vertex of the parabola is at the ordered pair, (h,k). If graphed, this parabola would open either up or down. The equation for a left or right opening parabola in vertex form would be written as x=a(y-k)

^{2}+h.

As you can see, the “a” value is a common element in both forms of the quadratic equation. This value holds the key to which direction the parabola will open.

If the “a” value is positive, equations with the x-variable squared will graph as parabolas that open UP. Equations with the y-variable squared will graph as parabolas that open RIGHT.

If the “a” value is negative, equations with the x-variable squared will graph as parabolas that open DOWN. Equations with the y-variable squared will graph as parabolas that open LEFT.

To put all of this information together, let’s look at a few examples of quadratic equations:

- 1. y=2×2+3x-6

This is a standard form quadratic equation with the x-variable squared and a=2. Because a>0, the parabola will open “up.”

- 2. y=-4(x-3)2+2

This is a vertex form quadratic equation with the x-variable squared, vertex at the ordered pair (3,2) and a=-4. Because a<0the parabola will open “down.”

- 3. x=3y2-2y-1

This is a standard form quadratic equation with the y-variable squared and a=3. Because a>0the parabola will open “right.”

- 4. x=-(y-2)2+1

This is a vertex form quadratic with the y-variable squared, vertex at the ordered pair (1,2) and a=-1. Because a<0, the parabola will open “left.”

Thanks for watching this review of parabolas! I hope it was helpful. See you next time!