Problem 36 - Precalculus: Mathematics for Calculus

Question

Graphing an Equation Using Rotation of Axes Show that the graph of the equation $$\sqrt{x}+\sqrt{y}=1$$ is part of a parabola by rotating the axes through an angle of \(45^{\circ} .\) [ Hint: First convert the equation to one that does not involve radicals. \(]\)

Step-by-Step Solution

Answer

Transform the equation using a rotation of 45 degrees; it forms a parabola equation.

Step 1: Convert Equation to Eliminate Radicals

The original equation is \( \sqrt{x} + \sqrt{y} = 1 \). Assume \( \sqrt{x} = u \) and \( \sqrt{y} = v \). Then the equation becomes \( u + v = 1 \). So we have \( x = u^2 \) and \( y = v^2 \).

Step 2: Express Terms in Terms of a Single Variable

Using the expressions for \( x \) and \( y \), the equation can be rewritten: \( u + v = 1 \), where \( u = 1 - v \). Substitute back to get \( x = (1-v)^2 \) and \( y = v^2 \).

Step 3: Relate Variables Using Parabola Equation

The relation \( x = (1-v)^2 \) and \( y = v^2 \) suggests: \( x = 1 - 2v + v^2 \). Hence, the equation of the relation can be represented as \( x = 1 - y + 2\sqrt{y} \).

Step 4: Rotate Coordinates by 45 Degrees

Rotate the coordinate system by 45 degrees using transformations: \( X = \frac{1}{\sqrt{2}}(x-y) \) and \( Y = \frac{1}{\sqrt{2}}(x+y) \). Substitute back to express the new equation in terms of \( X \) and \( Y \).

Step 5: Simplify and Identify Parabola Equation

Using the transformations, substitute and simplify to match standard parabola forms. Expressions will reduce to a form resembling \( (X+a)^2 = 4pY \), which is the equation of a parabola.

Key Concepts

Graphing Equations

Graphing equations can be a fascinating process that involves understanding the behavior and shapes of mathematical relationships on a coordinate system. Given our original equation \( \sqrt{x} + \sqrt{y} = 1 \), our task is to visualize its graph.

To graph this effectively, we first need to transform our equation. By setting \( \sqrt{x} = u \) and \( \sqrt{y} = v \), we eliminate the square root symbols, simplifying our work. This substitution converts the equation to \( u + v = 1 \), indicating a direct relationship between the transformed variables.

This step is crucial because it simplifies the complexity of the radicals and allows us to express \( x \) and \( y \) in terms of \( u \) and \( v \), as \( x = u^2 \) and \( y = v^2 \). Graphing in this form typically showcases how changes in one variable affect the overall interaction between them, helping create a more comprehensive graph.

Parabola

Parabolas are U-shaped curves that are essential in the world of graphs. They represent the set of all points that satisfy a quadratic equation. In our context, we aim to transform the equation \( \sqrt{x} + \sqrt{y} = 1 \) such that it takes the form of a parabola equation.

Once we have \( x = (1-v)^2 \) and \( y = v^2 \), we can express \( x \) as \( x = 1 - 2v + v^2 \). This showcases a squared term inherent in parabolic equations, indicating that our transformation is successful. The standard form of a parabola is \( (X+a)^2 = 4pY \).

Recognizing this pattern allows us to comprehend how our transformed equation behaves like a parabola. By comparing it with the standard quadratic form, we identify the 'vertex' and orientation of our parabola. Such visualization is crucial to understanding how parabolas work and how they represent real-world phenomena like paths of projectiles or satellite dishes.

Coordinate Transformation

Coordinate transformation is a process where we change the perspective or orientation of the coordinate system to simplify equations. For our equation, we rotate the axes by \( 45^{\circ} \) to reveal its parabolic nature more explicitly.

We use transformation equations: \( X = \frac{1}{\sqrt{2}}(x-y) \) and \( Y = \frac{1}{\sqrt{2}}(x+y) \). This rotation is known as a linear transformation because it keeps the system's linearity intact but changes the layout.

Implementing this transformation reveals the new relationships between \( X \) and \( Y \), simplifying the expression to potentially match the parabolic form. This allows us to identify the resemblance to \( (X+a)^2 = 4pY \). Such transformations make it easier to understand complex equations and graphical forms, showing the power of analyzing from different angles and orientations.