Graph Of X 2y 6

Exploring the Graph of x² + 2y = 6: A Comprehensive Guide

This article provides a comprehensive exploration of the graph represented by the equation x² + 2y = 6. We will delve into its characteristics, analyze its shape, and understand its behavior in the Cartesian coordinate system. This exploration will cover methods for plotting the graph, its classification within conic sections, and answer frequently asked questions about its properties. Understanding this seemingly simple equation opens doors to a deeper understanding of algebraic representation and geometrical visualization.

Understanding the Equation: x² + 2y = 6

The equation x² + 2y = 6 represents a parabola. It's a quadratic equation in one variable (x) and linear in the other (y). This specific form allows us to easily manipulate the equation to highlight its key properties and aid in plotting. Let's rearrange it to solve for y:

2y = 6 - x²

y = 3 - (1/2)x²

This form clearly shows that the equation represents a parabola that opens downwards. The negative coefficient of the x² term dictates the downward opening direction. The vertex and other key features can be easily derived from this form.

Plotting the Graph: Step-by-Step Approach

Plotting the graph involves identifying key points and connecting them to form the parabolic curve. Here’s a step-by-step approach:

1. Finding the Vertex: The vertex of a parabola in the form y = ax² + bx + c is given by x = -b/2a. In our case, y = -(1/2)x² + 0x + 3, so a = -1/2, b = 0, and c = 3. Therefore, the x-coordinate of the vertex is x = -0/(2*(-1/2)) = 0. Substituting x = 0 into the equation gives y = 3. Thus, the vertex is at (0, 3).

2. Finding the x-intercepts: The x-intercepts are the points where the graph intersects the x-axis (where y = 0). Setting y = 0 in the equation:

   0 = 3 - (1/2)x²

   (1/2)x² = 3

   x² = 6

   x = ±√6

Therefore, the x-intercepts are approximately (-2.45, 0) and (2.45, 0).

3. Finding the y-intercept: The y-intercept is the point where the graph intersects the y-axis (where x = 0). Setting x = 0 in the equation:

   y = 3 - (1/2)(0)² = 3

Therefore, the y-intercept is (0, 3), which is also the vertex.

4. Plotting Additional Points: To get a more accurate representation of the parabola, we can calculate a few more points by substituting different x-values into the equation and solving for y. For example:

   • If x = 1, y = 3 - (1/2)(1)² = 2.5
   • If x = -1, y = 3 - (1/2)(-1)² = 2.5
   • If x = 2, y = 3 - (1/2)(2)² = 1
   • If x = -2, y = 3 - (1/2)(-2)² = 1

5. Drawing the Parabola: Plot the vertex, x-intercepts, y-intercept, and the additional points you calculated on a Cartesian coordinate system. Connect the points smoothly to form a downward-opening parabola.

Analyzing the Graph: Key Features and Properties

The graph of x² + 2y = 6 exhibits several key features characteristic of a parabola:

• Axis of Symmetry: The parabola is symmetric about the y-axis (x = 0). This is because the equation is a function of x². Any point (x, y) on the parabola has a corresponding point (-x, y) also on the parabola.

• Vertex: The vertex, as calculated earlier, is at (0, 3). This is the highest point on the parabola.

• Concavity: The parabola opens downwards due to the negative coefficient of the x² term (-1/2).

• Focus and Directrix: While not directly apparent from the given equation, the parabola has a focus (a point) and a directrix (a line). The distance from any point on the parabola to the focus is equal to the distance from that point to the directrix. These can be calculated using the standard form of a parabola equation, but are beyond the scope of this basic explanation.

• Domain and Range: The domain (all possible x-values) is all real numbers (-∞, ∞). The range (all possible y-values) is (-∞, 3].

The Parabola in the Context of Conic Sections

The equation x² + 2y = 6 belongs to the family of conic sections. Conic sections are curves obtained by intersecting a cone with a plane. A parabola is one type of conic section, characterized by its single focus and directrix. Other conic sections include circles, ellipses, and hyperbolas. The equation's form distinguishes it as a parabola rather than another type of conic section.

Further Exploration and Applications

The understanding of parabolic graphs is crucial in numerous fields:

• Physics: The trajectory of a projectile under the influence of gravity follows a parabolic path.

• Engineering: Parabolic reflectors are used in antennas and satellite dishes to focus signals.

• Mathematics: Parabolas are fundamental in calculus, particularly in optimization problems.

• Computer Graphics: Understanding parabolic curves is essential for creating realistic 3D models and animations.

Frequently Asked Questions (FAQ)

Q1: Can this equation be expressed in a different form?

A1: Yes, the equation can be expressed in various forms. We already saw the solved form y = 3 - (1/2)x². It could also be expressed in the standard form of a parabola (x - h)² = 4p(y - k), where (h, k) is the vertex and p is the distance from the vertex to the focus. Finding the standard form requires some algebraic manipulation.

Q2: What is the significance of the coefficient -1/2 in the equation?

A2: The coefficient -1/2 determines the parabola's “width” and the direction it opens. A smaller absolute value of this coefficient leads to a wider parabola. The negative sign indicates that the parabola opens downwards.

Q3: How can I find the focus and directrix of this parabola?

A3: This requires converting the equation into the standard form (x - h)² = 4p(y - k). Once in this form, the focus is at (h, k + p) and the directrix is the horizontal line y = k - p. This involves completing the square and involves a more advanced mathematical process.

Q4: Are there other methods to plot this parabola besides the point-plotting method?

A4: Yes, you can use graphing calculators or software to plot the parabola directly by inputting the equation. These tools often provide additional information about the graph's properties.

Conclusion

The graph of x² + 2y = 6 represents a downward-opening parabola with vertex at (0, 3). Understanding its equation allows us to easily determine its key features, such as its vertex, x-intercepts, y-intercept, axis of symmetry, and concavity. By plotting key points and connecting them smoothly, we can accurately visualize the parabola. This seemingly simple equation serves as a valuable example illustrating the connection between algebraic representation and geometric visualization, a fundamental concept in mathematics and its numerous applications across various fields. Further exploration into the concepts of focus, directrix, and conic sections provides a deeper understanding of its properties within the broader context of mathematics.