Problem 65 - Calculus of a Single Variable

Question

Sketching a Graph In Exercises \(59-74\) , sketch the graph of the equation using extrema, intercepts, symmetry, and asymptotes. Then use a graphing utility to verify your result. $$ x y^{2}=9 $$

Step-by-Step Solution

Answer

The graph of the equation \(x y^{2}=9\) is a hyperbola. It has x-intercept at x=0. It does not intersect the y-axis. The graph has symmetry about the origin and has no asymptotes.

Step 1: Identify the equation type

The equation \(x y^{2}=9\) is a hyperbola because it is a quadratic equation in two variables.

Step 2: Find the intercepts

The intercepts of the equation can be found by setting one variable equal to zero and solving for the other variable. For the x-intercept, set \(y=0\) and solve for \(x\). It gives \(x=0\). For the y-intercept, set \(x=0\) and the equation becomes undefined. Hence, the graph does not intersect the y-axis.

Step 3: Check for symmetry

To identify symmetry, replace \(x\) with \(-x\) and \(y\) with \(-y\) in the equation, then simplify. If the equation remains unchanged, then it has symmetry. This equation does not change when both \(x\) and \(y\) are replaced with their negatives, so it has symmetry about the origin.

Step 4: Determine the asymptotes

For a hyperbola, asymptotes may be identified by setting the equation equal to zero. The equation cannot be rearranged into the form \((y-k)^2/a^2 - (x-h)^2/b^2 = 1\) or \((x-h)^2/a^2 - (y-k)^2/b^2 = 1\). Thus, there are no asymptotes.

Step 5: Verification

Verify your result by comparing your sketch with the one obtained from a graphing utility

Key Concepts

Intercepts

In any graph sketching exercise, finding intercepts is pivotal to understand where the curve meets the axes. For the equation \(x y^{2}=9\), finding intercepts involves setting either \(x\) or \(y\) to zero.
To determine the x-intercept, set \(y = 0\). However, this leads to \(x = 0\), which means that the graph covers the point (0,0) on the x-axis.
Besides, when trying to find the y-intercept by setting \(x = 0\), the equation becomes \(0 \cdot y^2 = 9\), which results in an undefined scenario. Thus, there is no y-intercept for this hyperbola.
**Key Points:**

• X-Intercept: The graph intersects the x-axis at the origin (0,0)
• Y-Intercept: The graph does not intersect the y-axis

Intercepts are significant as they help anchor the graph clearly on the Cartesian coordinate system.

Symmetry

Symmetry in graphs can simplify how you sketch the curve since symmetric shapes reflect across certain axes or points.
For the equation \(x y^{2}=9\), checking symmetry involves replacing \(x\) with \(-x\) and \(y\) with \(-y\). If the equation remains unchanged, it possesses symmetry.
In this instance, replacing gives \(x(-y)^2 = 9\), which simplifies back to \(x y^{2} = 9\). This confirms the graph is symmetric with respect to the origin.
**Symmetric Characteristics:**

• The equation does not change when both \(x\) and \(y\) are replaced with their negatives
• Indicates symmetry about the origin

Understanding symmetry is a powerful tool in predicting the behavior of graphs away from just calculated points.

Hyperbola

The equation \(x y^{2}=9\) describes a specific type of graph known as a hyperbola. Hyperbolas are curves formed by intersecting a double cone.
**Hyperbola Basics:**

• Typically represented as \(\frac{(x-h)^{2}}{a^{2}} - \frac{(y-k)^{2}}{b^{2}} = 1\) (or vice versa)
• Shows two branches that ‘mirror’ each other

However, our particular equation does not conform directly to this standard form, yet the principle of a hyperbola with separate branches still applies.
**Visualizing our Equation:**

• The graph has two distinct branches, mainly due to variable exponents
• By reshaping the equation for specific \(x\) values, you can trace the typical hyperbola arc

Recognizing the hyperbolic nature in equations helps in anticipating how the curve extends and behaves.

Asymptotes

Asymptotes are lines that a curve approaches as it heads towards infinity, but never really touches. They inform about a graph's behavior at extreme values.
For our equation \(x y^{2}=9\), identifying standard asymptotes can be tricky. In conventional hyperbolas, asymptotes are determined by setting an equation form to zero.
However, this equation lacks the typical hyperbola's format \((y-k)^2/a^2 - (x-h)^2/b^2 = 1\), making asymptote derivation not possible via easy manipulation.
**Asymptote Summary:**

• In this case, the equation does not lend itself to clear asymptote determination
• Simpler visual aids might suggest where values approach indefinitely

Asymptotes drive how a hyperbola expands without intersecting these invisible boundary guides.