Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 10: Problem 29
Graph the equation. $$\sqrt{2} x^{2}+2 \sqrt{2} x y+\sqrt{2} y^{2}-8 x+8 y=0$$
Short Answer
Expert verified
The equation represents a rotated parabola given by \( (x + y)^{2} = 8(x - y) \).
Step by step solution
01
- Rewrite the equation
Rewrite the given equation to separate the quadratic and linear terms. The original equation is: \(\[\begin{equation}\begin{aligned}\sqrt{2} x^{2}+2 \sqrt{2} x y+\sqrt{2} y^{2}-8 x+8 y=0 \end{aligned}\end{equation}\] \)
02
- Factor out constants
Factor out \( \sqrt{2} \) from each term in the equation:\( \sqrt{2}( x^{2} + 2xy + y^{2}) - 8x + 8y = 0 \)
03
- Complete the square
We rewrite the quadratic part in a complete square form. The quadratic expression \( x^{2} + 2xy + y^{2} \) can be factored as: \( (x + y)^{2} \)So the equation becomes: \( \sqrt{2}( (x + y)^{2} ) - 8x + 8y = 0 \)
04
- Re-organize the equation
Next, rearrange the terms to simplify the expression: \(( x + y)^{2} = \sqrt{2}(8x - 8y) \)
05
- Simplify the equation
Combine and simplify the terms: \(( x + y)^{2} = 8(x - y) \)
06
- Recognize the graph form
The equation \( (x + y)^{2} = 8(x - y) \) represents a rotated parabola. To interpret it on the Cartesian plane, convert it to standard form for conic sections.
07
- Conic form analysis
Observe that the equation combines both quadratic terms \( (x + y)^{2} \) and linear terms. This characterizes a rotated parabola opening asymmetrically due to the combined terms transforming a standard form quickly visible factors as geometric understanding.
08
- Graph the equation
Using a graphing utility or by plotting specific points, sketch the rotated parabola based on the equation \( (x + y)^{2} = 8(x - y) \). It opens upwards along a transformed axis line.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Completing the Square
Completing the square is a method used to transform a quadratic equation into a perfect square trinomial. This is particularly useful when working with conic sections or when solving quadratic equations. In our problem, the equation \( x^{2} + 2xy + y^{2} \) can be rewritten as (x + y)^{2}. This step helps to simplify the equation and make it easier to graph.
Completing the square involves the following steps:
Completing the square involves the following steps:
- 1. Take the quadratic part of the equation.
- 2. Split it into a form that represents a squared binomial.
- 3. Balance the equation by making any necessary adjustments.
Quadratic Equations
Quadratic equations are polynomial equations of the form ax^2 + bx + c = 0, where a, b, and c are constants. They represent parabolas in coordinate geometry. Our given equation \( \sqrt{2} x^{2}+2 \sqrt{2} x y+\sqrt{2} y^{2}-8 x+8 y=0 \), includes not only quadratic terms in x and y but also linear terms and a mixed xy term.
To solve or graph a quadratic equation:
To solve or graph a quadratic equation:
- 1. Look to factorize or simplify the equation.
- 2. Use methods like completing the square or the quadratic formula.
- 3. Analyze the equation's geometric properties such as vertex, focus, and axis of symmetry.
Conic Sections
Conic sections are curves that can be formed by the intersection of a cone and a plane. These include circles, ellipses, parabolas, and hyperbolas. They play a vital role in coordinate geometry.
In our example, the given equation transforms into a rotated parabola \( (x + y)^{2} = 8 (x - y) \). This is not a standard form parabola as it is rotated.
Conic sections can be categorized and understood by:
In our example, the given equation transforms into a rotated parabola \( (x + y)^{2} = 8 (x - y) \). This is not a standard form parabola as it is rotated.
Conic sections can be categorized and understood by:
- Recognizing the standard forms (e.g., x^2 + y^2 = r^2 for circles).
- Using transformations and rotations to understand more complex forms.
- Analyzing key properties such as focus, directrix, and eccentricity.
Coordinate Geometry
Coordinate geometry involves representing geometric figures like lines, circles, parabolas, etc., on a coordinate plane. It combines algebra and geometry and gives a dynamic way to explore shapes and their properties.
In our exercise, coordinate geometry helps us place the equation \( (x + y)^{2} = 8(x - y) \) on the Cartesian plane. This graph represents a rotated parabola.
To graph these complex curves, one should follow these steps:
In our exercise, coordinate geometry helps us place the equation \( (x + y)^{2} = 8(x - y) \) on the Cartesian plane. This graph represents a rotated parabola.
To graph these complex curves, one should follow these steps:
- 1. Transform the equation into a more recognizable form.
- 2. Identify key features like vertex, axis of symmetry, and directrix.
- 3. Plot points to understand the spread and direction of the curve.
