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Magazine Chapter 3: Problem 45
Plot the Curves : $$ x^{2} y^{2}+y=1 $$
Short Answer
Expert verified
Plot by finding points satisfying \( x^2 y^2 + y = 1 \) and using symmetry.
Step by step solution
01
Understand the Equation
The given equation is \( x^2 y^2 + y = 1 \). This represents a curve in the \( xy \)-plane. Our task is to plot this curve by analyzing the equation for values of \( x \) and \( y \) that satisfy the equation.
02
Express y in terms of x
Rewrite the equation as \( y(x^2 y + 1) = 1 \). Solving for \( y \), we get \( y = \frac{1}{x^2 y + 1} \). This expression must hold true for all points \((x, y)\) on the curve.
03
Analyze Symmetry and Domain
Since both \( x \) and \( y \) appear symmetrically in the equation, the curve is symmetric about the \( y \)-axis and the \( x \)-axis. The domain of \( x \) is all real numbers, but \( x^2 y + 1 \) cannot be zero, as it would make \( y \) undefined.
04
Explore Particular Points
Set \( x = 0 \), we get \( y = 1 \). At \( y = 0 \), there is no solution because the left-hand equation would be zero, which does not satisfy \( x^2 y^2 + y = 1 \). Try some fixed values: \( x=1 \) gives a quartic equation in \( y \). Similarly, try other simple values such as \( x = -1 \), \( x = \sqrt{2} \), etc.
05
Plot Using Points and Symmetry
Use a graphing tool to plot points for each calculated \( (x, y) \) pair. Utilize the symmetry of the graph about the \( x \)-axis and \( y \)-axis to plot additional points. Connect these points smoothly to form the curve.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Symmetry in Equations
When working with equations like \( x^2 y^2 + y = 1 \), identifying symmetry helps us understand the shape and features of the curve without plotting every single point. In this equation, both \( x \) and \( y \) are involved in squared terms, which indicates potential symmetry with respect to the axes.
Symmetries observed:
Symmetries observed:
- **Symmetry about the x-axis**: If \((x, y)\) is a point on the curve, then \((x, -y)\) should also satisfy the equation, promoting symmetry relative to the x-axis.
- **Symmetry about the y-axis**: Similarly, if \((-x, y)\) satisfies the equation for a positive x-value, this reveals symmetry about the y-axis.
Expression Manipulation
Expression manipulation is a handy tool that involves rewriting equations to make them easier to analyze or solve. In the exercise \( x^2 y^2 + y = 1 \), rearranging terms can reveal more about potential solutions.
**How to manipulate the expression**
**How to manipulate the expression**
- Start by isolating \( y \): \( y(x^2 y + 1) = 1 \). This intermediate form highlights that the possible values of \( y \) are dependent on \( x^2 \).
- Express \( y \) explicitly: \( y = \frac{1}{x^2 y + 1} \).
Domain and Range
Domain and range help determine the scope of x and y values for which the equation has a solution. With the curve \( x^2 y^2 + y = 1 \), it's essential to understand both aspects fully.
**Determining domain**:
**Determining domain**:
- The domain is concerned with permissible values of \( x \), which is typically all real numbers. However, check the condition \( x^2 y + 1 eq 0 \), as this could make \( y \) undefined.
- The range involves all potential \( y \) values. Begin by evaluating special scenarios such as x = 0, yielding \( y = 1 \), yet understand the equation poses restrictions in other x selections (e.g., there is no solution when \( y = 0 \)).
Graphical Analysis
Graphical analysis is the process of understanding equations by looking at them visually to decipher their behavior. For \( x^2 y^2 + y = 1 \), visualizing involves both drawing and investigating the curve's behavior.
**Steps for effective graphical analysis**:
**Steps for effective graphical analysis**:
- Choose critical points and symmetry: Start plotting points using symmetry to reduce effort. Use known symmetries to anticipate the resting position of reflective points to streamline plotting.
- Apply computational analysis: Calculate specific values at chosen \( x \) to understand various \( y \) outputs. Utilization of values like \( x = 0 \) or \( x = 1 \) is recommended for simplicity.
- Connect points to form the curve: Use these guides to construct the overall picture using smooth lines to connect plotted points, respecting symmetry and domain limitations.
