Problem 33 - Algebra 2

Question

Write each equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hyperbola. Then graph the equation. $$ x^{2}+2 y^{2}=2 x+8 $$

Step-by-Step Solution

Answer

The equation is an ellipse.

Step 1: Rearrange the Terms

Start by moving the linear term involving \(x\) to the left side by subtracting \(2x\) from both sides to get \(x^2 - 2x + 2y^2 = 8\).

Step 2: Complete the Square for the x Terms

To complete the square for the \(x\) terms, take half of the coefficient of \(x\), square it, and add inside the equation: \(-2/2 = -1\) and \((-1)^2 = 1\). Therefore, add and subtract 1: \((x^2 - 2x + 1) + 2y^2 = 8 + 1\).

Step 3: Simplify and Recognize the Equation Form

The equation becomes \((x - 1)^2 + 2y^2 = 9\). Divide through by 9 to express it in a recognizable conic form: \(\frac{(x-1)^2}{9} + \frac{y^2}{4.5} = 1\).

Step 4: Identify the Conic Section

Since the equation is of the form \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\) and both \(a^2\) and \(b^2\) are positive, the graph is an ellipse.

Step 5: Graph the Equation

Identify the center of the ellipse at point \((1,0)\). The horizontal axis (\(a\)) has a length of 3 (since \(9 = 3^2\)), and the vertical axis (\(b\)) has a length of approximately 2.12 (since \(4.5 = b^2\), \(b \approx 2.12\)). Draw the ellipse centered at \((1,0)\) with these axes.

Key Concepts

Ellipse

An ellipse is a type of conic section formed by the intersection of a plane and a cone. It resembles an elongated circle and is defined by two main axes: the major axis (longest diameter) and the minor axis (shortest diameter).

When you encounter equations like the one given, it helps to determine if the shape is an ellipse by looking at the terms and their coefficients. An ellipse has the equation of the form: \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]where:

• The center is at point "(h, k)"
• "a" is the semi-major axis length
• "b" is the semi-minor axis length

If both "a" and "b" values are positive and not equal, you indeed have an ellipse.

In this case, the given equation is transformed to fit this mold and verify its nature as an ellipse. Information like this can be essential in identifying which type of conic section you're dealing with.

Standard form of equation

The standard form of an ellipse equation is handy for quickly identifying the essential characteristics of the ellipse. To convert an equation into this form, follow the precedence of rearranging and simplifying terms.

For equations involving both x and y, and which need to be simplified, you proceed with arranging terms in such a way that you can complete the square. This will allow for easy recognition of the form: \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]where each part links clearly to geometrical properties, like centering on the ellipse. From our original equation, "x^2 - 2x + 2y^2 = 8" , dissecting it into parts and managing constants results in a clear, interpretable standard form. The key here is identifying and rearranging with precision so that each element of the equation reflects the ellipse's properties: center, axes, and orientation.

Completing the square

Completing the square is pivotal in transforming a quadratic equation toward a more manageable form, which allows us to identify essential features like center and axes of an ellipse in our context.

This technique involves the following steps:

• Move terms to one side of the equation to isolate the quadratic terms first.
• Take half of the linear coefficient, square it, and adjust the equation accordingly by adding and subtract this square in the equation.

For example, in "x^2 - 2x + 2y^2 = 8", we manage the x terms by taking half of -2 (which is -1), squaring, hence inserting 1, turning it eventually into: "(x-1)^2".

This method simplifies our understanding visually on the graph, leading directly toward expressing in standard form,

Graphing equations

Graphing equations, especially for conic sections like ellipses, is about clear visual translation of mathematical expressions to the coordinate plane. This involves identifying key attributes from the equation in its standard form.

The steps include:

• Determine the center, typically indicated by "(h,k)" from the equation.
• Assess the lengths of the semi-major and semi-minor axes ("a" and "b"), which guide the ellipse's stretch direction and roundness.
• Plot the major and minor axis relative to the center.

Following our example, from the rearranged equation "(x-1)^2/9 + y^2/4.5 = 1", the graph centers at "(1,0)" with a horizontal reach of 3 units and vertical reach approximately 2.12 units.

This step-by-step transfer of algebra to geometry enhances understanding by offering concrete visual interpretation of abstract numeric relationships.