Problem 30 - Algebra and Trigonometry

Question

In Exercises 17–30, find the standard form of the equation of each parabola satisfying the given conditions. Focus: \((7,-1) ;\) Directrix: \(y=-9\)

Step-by-Step Solution

Answer

The standard form of the equation of the parabola is \((x-7)^2 = 16(y+5)\)

Step 1: Identifying Vertex

Since the directrix is horizontal (y = -9), the parabola opens upwards or downwards. Because the focus is above the directrix, this parabola opens upwards. The vertex of the parabola is midway between the focus and the directrix. Average the y-value of the focus and directrix to find the y-value of the vertex: \(-1 + (-9)\) / 2 = -5. The x-value of the vertex is just the x-value of the focus which is 7. Therefore, the vertex of the parabola is (7,-5).

Step 2: Finding the Value of a

The value of \( a \) is the distance from the vertex to the focus (or to the directrix). As you have the vertex (7,-5) and focus (7,-1), you may subtract the y-values to find the value of \( a : -1 - (-5) = 4 \). Thus, \( a = 4 \).

Step 3: Writing the Equation of the Parabola

Substitute the vertex (h, k) and \( a \) into the vertex form of a parabola that opens upwards or downwards: \((x-h)^2 = 4a(y-k)\). This gives: \((x-7)^2 = 4*4(y+5)\). Simplifying this results in \((x-7)^2 = 16(y+5)\). Thus, this is the standard form of the equation of the parabola that satisfies the given conditions.

Key Concepts

Vertex

The vertex of a parabola plays a crucial role in understanding its shape and position on the coordinate plane. In the context of a parabola, the vertex is the point where the curve makes its sharpest turn. It is essentially the 'tip' of the parabola if it opens upwards or downwards, or the side point if it opens left or right.
To pinpoint the vertex, you need to consider both the focus and the directrix. The vertex is located halfway between the focus and the directrix. In our example with a focus at (7, -1) and a directrix at \(y = -9\), the y-value of the vertex is calculated as the average:

• \((-1 + (-9)) / 2 = -5\)

The x-coordinate of the vertex aligns with the x-coordinate of the focus, so:

• The vertex here is (7, -5).

Recognizing the vertex is the first step to writing the parabola's equation.

Directrix

A directrix of a parabola is a crucial component that helps define its shape. It is a fixed line that, together with the focus, influences the position of the parabola.

• For parabolas opening upwards or downwards, the directrix is a horizontal line; for those opening left or right, it's vertical.

In this example, the directrix is given by the equation \(y = -9\), illustrating that the parabola opens upwards or downwards.
The vertex of the parabola will lie somewhere vertically between the directrix and focus. Given the focus at \((7, -1)\), the vertex at \((7, -5)\) confirms that the opening is upwards since the focus is above the directrix.
Understanding the directrix gives insight into how the parabola is oriented and helps determine the equation form.

Focus

The focus of a parabola acts as one of the key characteristics defining its structure. It is a fixed point that, along with the directrix, guides the formation of the parabola.

• The parabola's curve reflects around the focus towards the directrix.

In our scenario, the focus is placed at \((7, -1)\). This placement above the directrix \(y = -9\) indicates that the parabola's opening is upwards. By calculating the midpoint between the focus and the directrix, we locate the vertex at \((7, -5)\). The distance between the vertex and the focus is crucial in determining \(a\), a value used in the parabola's equation.
By understanding the focus's position, one can discern how the parabola will be shaped and oriented on the plane.

Standard Form of Parabola

The standard form of a parabola's equation provides a functional representation of its curve. This form can give insights into the vertex, axis of symmetry, and direction of opening.
For a parabola with its axis of symmetry parallel to the y-axis, the standard form is \((x-h)^2 = 4a(y-k)\), where:

• \((h, k)\) is the vertex,
• \(a\) is the distance between the vertex and the focus or directrix.

In our case, with a vertex at \((7, -5)\) and focus at \((7, -1)\), \(a\) equals 4. Plugging these values into the standard form gives:

• \((x-7)^2 = 16(y+5)\)

This equation represents the parabola in standard form, showcasing its upward opening, vertex position, and curved nature. Understanding how to transition from the vertex and focus to this equation aids in visualizing and analyzing the parabola's characteristics.