Problem 54 - Precalculus

Question

Find the standard form of the equation of the parabola with the given characteristics. Vertex: (1,2)\(;\) directrix: \(y=-1\)

Step-by-Step Solution

Answer

The standard form of the equation for the parabola with vertex at (1,2) and directrix at \(y=-1\) is \(y=(1/12)*(x-1)^2 + 2\).

Step 1: Find the value of p

From the problem, we know that the vertex is at (1, 2) and the directrix is \(y=-1\). The vertical distance from a point to a line is simply the absolute difference in their \(y\) coordinates. So, the distance \(p\) from the vertex \((1,2)\) to the directrix line \(y=-1\) is \(p = abs(2--1) = 3\).

Step 2: Find the value of a

We find the parameter \(a\) using the formula \(a=1/(4p)\) We substitute the value of \(p = 3\) into the formula to get \(a = 1/12\).

Step 3: Write the parabola equation in standard form

We can now write our parabola in vertex form: \(y=a(x-h)^2 +k\). Substituting our values we have found we get \(y=(1/12)*(x-1)^2 + 2\). This is the standard form of the parabola which satisfy the given conditions.

Key Concepts

Vertex Form of a Parabola

The vertex form of a parabola's equation is a very useful representation, especially when you already know the vertex of the parabola. The general formula is given by:

• \( y = a(x-h)^2 + k \)

Here, \((h, k)\) represents the vertex of the parabola. This form is beneficial because it immediately tells us the highest or lowest point on the parabola, making it easier to graph.
The "\(a\)" value in the equation influences the width and direction of the parabola. If \(a > 0\), the parabola opens upwards, and if \(a < 0\), it opens downwards.
In the context of the original exercise, the vertex was provided as \((1,2)\), leading to an equation of the form \( y = a(x-1)^2 + 2 \). From this, it is clear how the vertex form is specifically calculated utilizing the vertex data.

Directrix of a Parabola

Understanding the directrix is key to understanding the geometry of parabolas. A directrix is a fixed line used in the formal definition of a parabola. Any point on the parabola is equidistant to a point called the 'focus' and the directrix line. This unique characteristic helps define the shape and position of the parabola on a coordinate plane.
In the exercise, the directrix is given as \(y = -1\). This tells us that the parabola is oriented vertically, as the vertex \((1,2)\) is above the directrix. The distance from the vertex to the directrix is termed \(p\), calculated using the vertical distance formula. Here we have \(p = 3\), helping us further to determine the parabola's shape and orientation by solving for "\(a\)" in the vertex form.

Standard Form of the Equation

While the vertex form is handy for identifying a parabola's vertex easily, the standard form is often used for calculations and analyses. The standard form of a parabola's equation is:

• \( ax^2 + bx + c = y \)

In this form, \(a\), \(b\), and \(c\) are constants that determine the parabola's shape and position. The standard form doesn't directly tell us the vertex; it requires a bit of calculation to derive that information.
In the solution provided, the vertex form \( y = (1/12)(x-1)^2 + 2\) is already simplified to align with the given characteristics, showing that sometimes the conversion from vertex form might not be essential for certain problems. Nevertheless, understanding both forms is crucial, as they offer different insights into the nature of parabolas.