Problem 75 - Intermediate Algebra

Question

Write each equation in standard form, if it is not already so, and graph it. If the graph is a circle, give the coordinates of its center and its radius. If the graph is a parabola, give the coordinates of its vertex. $$ x=-6(y-1)^{2}+3 $$

Step-by-Step Solution

Answer

The graph is a parabola with vertex at (3, 1), opening left.

Step 1: Identify the Type of Equation

The given equation is \( x = -6(y-1)^2 + 3 \). This equation is a conic section, specifically a parabola. This is evident because it is based on the standard form of a parabolic equation \( x = a(y-k)^2 + h \) which opens either left or right.

Step 2: Convert to Standard Form

The equation \( x = -6(y-1)^2 + 3 \) is already in the standard form for a parabola that opens horizontally: \( x = a(y-k)^2 + h \). In this form, \( a = -6 \), \( h = 3 \), and \( k = 1 \).

Step 3: Determine the Vertex

For the given equation \( x = -6(y-1)^2 + 3 \), extracted from the standard form, the vertex is \((h, k) \). Hence, the vertex of the parabola is \( (3, 1) \).

Step 4: Determine the Direction of Opening

Since \( a = -6 \) is negative, the parabola opens to the left. This indicates that as \( y \) values move away from the vertex, the \( x \) values decrease.

Step 5: Graph the Parabola

To graph the equation, plot the vertex at \( (3, 1) \) on a coordinate plane. Since the parabola opens left, plot a few points by choosing \( y \) values around \( y = 1 \) (like \( y = 0 \) and \( y = 2 \)) and calculate corresponding \( x \) values. Draw a smooth curve through these points extending leftward from the vertex.

Key Concepts

Conic Sections

In mathematics, conic sections are the curves obtained by intersecting a plane with a double-napped cone. These curves have significant applications in various fields like physics, engineering, and astronomy. The primary types of conic sections are **circles**, **ellipses**, **parabolas**, and **hyperbolas**. Each of these has unique properties and equations.

A key feature of conic sections is that their equations can be expressed in a general quadratic form:\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]The values of the coefficients (A, B, C, etc.) determine the type of conic shape. When dealing with a parabolic equation, as in the example \( x = -6(y-1)^2 + 3 \) given in the exercise, it's important to recognize that this specific equation describes a parabola, a type of conic section characterized by its U-shaped curve.

Parabolas have distinct behaviors compared to the other conic sections, primarily because each has a vertex that acts as the focal point of symmetry for the curve.

Graphing Parabolas

Graphing parabolas involves understanding the equation of the parabola and the position of its key features, such as the vertex and the direction in which it opens. The standard form of a parabolic equation is either \[ y = a(x-h)^2 + k \]for a vertically oriented parabola or \[ x = a(y-k)^2 + h \]for a horizontally oriented parabola, like the one in the exercise.

To graph the parabola from the exercise equation \( x = -6(y-1)^2 + 3 \):

• Identify the vertex using the components \( h \) and \( k \) from the equation. In this case, the vertex is \( (3, 1) \).
• Determine the direction of opening by examining the coefficient \( a \). Here, \( a = -6 \) indicates that the parabola opens to the left because a negative \( a \) means the parabola opens opposite to the standard direction.
• Choose several \( y \) values around the vertex (e.g., \( y = 0, 1, 2 \)) and calculate the corresponding \( x \) values to find other points on the parabola.
• Draw a smooth curve through the vertex and these additional points, ensuring the direction corresponds with the coefficient \( a \).

With these steps, you can accurately plot the parabola on a graph, allowing for better visualization of its shape and properties.

Vertex of a Parabola

The vertex of a parabola is a critical component in its graph, representing the point where the curve changes direction. The vertex can either be the highest or lowest point on the parabola, depending on its orientation.

In standard form, the vertex is denoted by the parameters \((h, k)\). For the equation \( x = -6(y-1)^2 + 3 \), the vertex is at \( (3, 1) \). This point not only acts as a pivot for the curve but also plays a significant role in determining other features of the parabola:

• **Axis of Symmetry**: The line passing through the vertex is known as the axis of symmetry, ensuring that one side of the parabola is a mirror image of the other.
• **Direction**: As mentioned before, the coefficient \( a \) dictates whether the parabola opens upward, downward, or horizontally. In our equation, the negative \(-6\) indicates a leftward opening.
• **Extremes**: Depending on the orientation, the vertex can indicate the minimum or maximum value of the parabola. For our horizontal parabola, it represents the boundary of change in direction along the \( y \)-axis.

Understanding the vertex's coordinates provides a foundation for analyzing and graphing parabolas, allowing you to predict and verify other characteristics of the parabola's curve.