Question

Which of the given characteristics describe parabolas that open down? Explain your reasoning. (A) focus: \((0,-6)\) directrix: \(y=6\) (B) focus: \((0,-2)\) directrix: \(y=2\) (C) focus: \((0,6)\) directrix: \(y=-6\) (D) focus: \((0,-1)\) directrix: \(y=1\)

Short answer

Characteristics (A), (B), and (D) describe parabolas that open downward.

Step-by-step solution

Identify the Correct Criteria for a Downward-Opening Parabola

Remember that for a downward-opening parabola, the y-coordinate of the focus is smaller than the y-coordinate of the directrix.

Analyze Option A

For option A, the focus is \((0,-6)\) and the directrix is \(y=6\). The y-coordinate of the focus is -6 which is smaller than 6, which is the y-coordinate of the directrix. Therefore, this parabola opens downward.

Analyze Option B

For option B, the focus is \((0,-2)\) and the directrix is \(y=2\). The y-coordinate of the focus is -2 which is smaller than 2, which is the y-coordinate of the directrix. Therefore, this parabola also opens downward.

Analyze Option C

For option C, the focus is \((0,6)\) and the directrix is \(y=-6\). The y-coordinate of the focus is 6 which is larger than -6, which is the y-coordinate of the directrix. Therefore, this parabola does not open downward.

Analyze Option D

For option D, the focus is \((0,-1)\) and the directrix is \(y=1\). The y-coordinate of the focus is -1 which is smaller than 1, which is the y-coordinate of the directrix. Therefore, this parabola also opens downward.

Key concepts

Focus

The focus of a parabola is a crucial component that helps us determine the parabola's orientation and shape. It is a fixed point located inside the parabola's curve. The unique property of any parabola is that every point on the curve is equidistant from the focus and the directrix.
This special property makes the focus meaningful in assessing the parabola's openness and direction. In downward-opening parabolas, the y-coordinate of the focus is always less than the y-coordinate of the directrix. This is because the parabola curves downward towards its vertex located between the focus and the directrix.

• For example, in question (A) where the focus is \(0, -6\), the low y-value of \-6\ helps the parabola to open downward, matching the criteria of a downward-opening orientation.
• Similarly, in example (B), with the focus \(0, -2\), the parabola curves down as the focus's y-coordinate also satisfies the requirement when compared to its directrix.

Observing the y-values of the focus in these examples is key to understanding the nature of parabolas.

Directrix

The directrix of a parabola is a line that, together with the focus, defines a parabola's shape and direction. It's important because it provides a boundary with which all points on the parabola maintain an equal distance when compared to the focus.
Specifically, the directrix is a line perpendicular to the axis of symmetry of the parabola. For downward-opening parabolas, the y-value of the directrix is always higher than that of the focus. This creates an upper boundary so that the parabola opens downwards.
In the examples given:

• In example (A), the directrix at \(y=6\) is positioned above the focus, indicating the parabola curves downwards.
• For option (B), the directrix \(y=2\) also sits above the focus, fulfilling the criteria for a downward-opening parabola.

Understanding the position of the directrix in relation to the focus gives insight into the parabola's orientation and the direction in which it opens.

Downward-Opening Parabolas

Downward-opening parabolas are specific types of parabolas that curve downwards to form a U-shape opening towards the negative y-axis. The defining characteristic of these parabolas is the relationship between the focus and directrix.
The principal condition for a parabola to open downward is that the focus lies below the directrix along the y-axis. This arrangement is necessary because such a configuration allows the vertex of the parabola to be positioned at a peak below the directrix, thus allowing the parabola to curve downwards.

• For instance, in both option (A) and option (B), the parabolas meet this condition and therefore open down, which is depicted by the smaller focus y-coordinate as compared to the y-coordinate of the directrix.
• However, in option (C), where the focus y-value is higher than that of the directrix, the parabola does not open downwards, illustrating that not every setup results in a downward curve.

Mastering the relationship between the focus and the directrix is fundamental to identifying the orientation of parabolas and predicting their behavior. This relationship dictates whether a parabola opens upwards or downwards in the Cartesian plane.