Problem 44 - Algebra for College Students

Question

Vertex \((0,0)\), focus \(\left(0,-\frac{7}{2}\right) \quad x^{2}=-14 y\)

Step-by-Step Solution

Answer

Vertex is (0,0) and focus is (0,-\frac{7}{2}). The equation correctly describes the parabola.

Step 1: Understanding the Equation

The given equation is \(x^{2} = -14y\), which is a standard form of a parabola that opens downwards. The general formula for a vertical parabola is \(x^{2} = 4py\), where \(p\) is the distance from the vertex to the focus.

Step 2: Finding the Value of p

To find \(p\), compare the equation \(x^{2} = -14y\) with the standard form \(x^{2} = 4py\). Here, we have \(4p = -14\). Solving for \(p\), we get: \[ p = \frac{-14}{4} = -\frac{7}{2} \].

Step 3: Locating the Focus

Since the vertex is at \((0, 0)\) and \(p = -\frac{7}{2}\), the focus is located \(p\) units below the vertex along the y-axis because the parabola opens downwards. Therefore, the focus is at \((0, -\frac{7}{2})\).

Step 4: Verifying the Focus

According to the problem, the focus is at \((0, -\frac{7}{2})\) which matches with our calculated position based on the value of \(p\). Therefore, the identified focus is correct.

Step 5: Conclusion

The equation \(x^{2} = -14y\) correctly describes a parabola with vertex \((0,0)\) and focus \((0, -\frac{7}{2})\) as given and analyzed.

Key Concepts

Parabolas

Parabolas are curved shapes that have a unique set of characteristics. If you imagine slicing a cone at just the right angle, the shape you get is a parabola. These curves are commonly seen in quadratic functions and are represented on a graph. They can either open upwards, downwards, leftwards, or rightwards, depending on the equation involved.

Here are a few important facts about parabolas:

• They have a singular curve that continuously rises or falls without any loops.
• The path of a thrown ball or the trajectory of an arrow often resembles a parabola.
• Mathematically, when given as a quadratic function in the form of \( y = ax^2 + bx + c \), you're dealing with a parabola.

Understanding parabolas is important because they show up in various real-world applications, from physics to architecture. Each parabola is defined by its vertex, focus, and the axis of symmetry, all of which determine how the graph looks and behaves.

Vertex

The vertex of a parabola is a special point where the curve changes direction. In simpler terms, it's either the lowest or highest point of the parabola, depending on whether it opens upwards or downwards. For the parabola given in the example, which is \( x^2 = -14y \), the vertex is at (0,0).

Key characteristics of the vertex include:

• It acts as the turning point of the parabola.
• For a parabola described by \( y = ax^2 + bx + c \), the vertex can be found using the formula \( x = \frac{-b}{2a} \) for the x-coordinate, while substituting this value back into the equation gives the y-coordinate.
• In a vertical parabola, it serves as a point equidistant to the focus and the directrix.

Remember, the vertex helps determine the symmetry and direction of the parabola, making it a crucial part of understanding these curves.

Focus

The focus of a parabola is another defining point located inside the curve. It is not on the parabola itself but is vital in determining the shape of the curve. In the exercise given, the focus is at \(\left(0, -\frac{7}{2}\right)\).

Here's what makes the focus important:

• It lies directly in line with the vertex along the axis of symmetry.
• The points on the parabola are equidistant to the focus and a line called the directrix.
• In the case of a downward-opening parabola, like \( x^2 = -14y \), the focus is found by moving down from the vertex by \( p \) units, where \( p \) is the distance from vertex to focus.

This relationship between the parabola, its focus, and the directrix defines the set of rules by which the curve is drawn, making the focus key to understanding how parabolas work.

Standard Form Equation

The standard form of a parabola's equation provides a simple yet comprehensive description of its geometry. For parabolas that open vertically, the equation is usually written as \( x^2 = 4py \), where \( p \) gives the distance from the vertex to the focus.

This form is particularly useful because:

• It easily shows whether the parabola opens up or down; a positive \( p \) means it opens upwards, while a negative \( p \) indicates a downward opening, as seen in \( x^2 = -14y \).
• It provides immediate insight into the parabola's axis of symmetry and vertex placement.
• By setting specific points like a vertex and focus, you can derive the standard form tailored to a particular parabola.

Understanding the standard form is necessary for analyzing the fundamental properties of parabolas, especially when connecting them to real-world problems or other mathematical functions.