#### From the Value of *a*

There are a couple of ways we can determine the direction the parabola opens. The a-value is one way. If a > 0, then the parabola opens upward. If a < 0, then the parabola opens downward. Recall that when the parabola is given in standard form, y = ax² + bx + c, the a-value is the leading coefficient.

**Example 1:**If y = 3x² -4x + 1, determine the parabola’s orientation.

**Solution:** In this example, a = 3, and because 3 > 0, the parabola opens upward. ■

The parabola can also be given in vertex form. Recall the vertex form of a parabola is given as y = a(x-h)² + k. We still use the a-value and use the same test when the equation of the parabola is given to us in vertex form.

**Example 2:**If y = -2(x+3)² – 3, determine the parabola’s orientation.

**Solution:** In this case, a = – 2 < 0. So this parabola opens downward. ■

#### From the Relative Position of the Directrix and Focus

Another way we can determine a parabola’s orientation is by knowing the *relative vertical position* between the focus and directrix. When the focus is above the directrix, then the parabola opens upward. When the focus is below the directrix, the parabola opens downward.

**Example:**A parabola’s focus is F(1,4) and directrix is y=7. Determine the parabola’s orientation.

**Solution:** Because the directrix is y = 7 and the y-coordinate of the focus is y = 4, the directrix is above the focus. Thus, the parabola opens downward. ■

#### From the Second Derivative and the Concavity Test

For you fancy-pants calculus students, you do not need to know anything about parabolas in general to know what direction it opens. Given the equation of the parabola, in either standard form or vertex form, you can perform a test for concavity by taking the second derivative of the given function. On a quadratic polynomial, the second derivative always yields a constant. When it yields a positive constant, we know the original function is everywhere concave up. When it yields a negative constant, the original function is everywhere concave down. In other words, when the second derivative of the equation of our parabola is positive, the parabola opens upward; when it is negative the parabola opens downward. That’s all there is to it. Here’s how it works in practice.

**Example 1:**If y = 2x² + 4x – 1, determine the parabola’s orientation.

**Solution:** Because y = 2x² + 4x – 1, then y’ = 4x + 4, and y”= 4. Because y” = 4 > 0, the parabola is everywhere concave up. Thus, the parabola opens upward. ■

Here is an example of determining the parabola’s orientation with the concavity test when the original equation is given in vertex form.

**Example 2:**If y = -2(x-1)² + 3, determine the parabola’s orientation.

**Solution:** If y = -2(x-1)² + 3, then y’ = -4(x-1), and y” = -4. Because y” = -4 < 0, the parabola is everywhere concave down. So the parabola opens downward. ■

Here’s what we learn when we apply the concavity test to the general quadratic polynomial.

**Example 3:**If y = ax² + bx + c, for a, b, and c real numbers, determine the parabola’s orientation.

**Solution:** Because y = ax² + bx + c, y’ = 2ax + b, and y”= 2a. Note that y” > 0 if a > 0, and y” < 0 if a < 0. So the value of a determines whether the parabola is everywhere concave up or everywhere concave down. Thus, the parabola opens upward whenever a > 0. The parabola opens downward whenever a < 0. ■

**Did we answer your question about parabola orientation? Share your thoughts in the comments.**