Problem 39 - Algebra 2

Question

Write an equation of a parabola with a vertex at the origin. focus at \((-7,0)\)

Step-by-Step Solution

Answer

The equation of the parabola is \(x = \frac{7}{4}y^2\)

Step 1: Identify the vertex and focus of the parabola

From the exercise, the vertex is at the origin (0,0) and the focus is at (-7,0).

Step 2: Determine the orientation of the parabola

Since the focus lies on the x-axis, and it is located to the left of the vertex, we can determine that the parabola opens towards the negative x-axis.

Step 3: Apply the formula for the standard form of a parabola

The standard form of a parabola that opens horizontally (to the left or right) is \(x = ay^2\). Our equation becomes \(x = ay^2\).

Step 4: Determine the value of \(a\)

The distance from the vertex to the focus is 7 units (the absolute value of the x-coordinate of the focus). The formula relating this distance to \(a\) in the standard form equation of the parabola is \(4a = 7\). So, \(a = 7/4\).

Step 5: Substitute the value of \(a\) into the standard form equation of the parabola

Substituting \(a = 7/4\) into the equation \(x = ay^2\), we get \(x = (7/4)y^2\).

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a critical point that defines its shape. In this exercise, the vertex is located at the origin, which is the point (0,0). This point can be seen as the starting point or anchor of the parabola. Depending on the parabola's orientation, the vertex represents either the lowest or highest point.

• If a parabola opens upwards (like a smile), the vertex provides its lowest point.
• If it opens downwards (like a frown), the vertex is its highest point.
• In a sideways-opening parabola, the vertex indicates the most leftward or rightward point. When the vertex of a parabola is at the origin, it simplifies calculations as it serves as a reference point for other properties of the parabola, like the focus or axis of symmetry.

Focus of a Parabola

The focus is a specific point used to define and construct a parabola. It gives the parabola its reflective property. In this exercise, the focus is positioned at (-7,0).

• The focal point directs the path of the parabola:
  Every point on a parabola is equidistant from the focus and a line called the directrix.
• The position of the focus determines the direction the parabola opens:
  In our case, the parabola opens towards the negative x-axis because the focus is left of the vertex.

This gives the parabola a horizontally opening shape rather than vertically. Understanding the location of the focus is crucial because it directly influences the shape and equation of the parabola.

Standard Form Equation of a Parabola

The standard form equation is crucial for understanding the structure of a parabola and its orientation. For parabolas that open sideways, the equation is typically given by \(x = ay^2\).

• The term \(a\) influences the parabola's width and direction:
  If \(a\) is positive, the parabola opens to the right; if negative, to the left.
• In this scenario, the value of \(a\) was found to be \(\frac{7}{4}\), derived from the distance between the vertex and the focus.
• Thus the equation becomes:
  \( x = \left( \frac{7}{4} \right)y^2 \)
  This equation describes a parabola that opens towards the left, consistent with the position of the focus at (-7,0).In the big picture, knowing the standard form equation helps in graphing the parabola and understanding its key features, like the width and direction of opening.