Problem 31 - Precalculus with Calculus Previews

Question

In Problems \(29-32\), use the discriminant to identify the conic. Rewrite the equation in the form given in (13) and find two functions defined implicitly by this equation. Give the domain of each function. Finally, use a graphing utility to graph these functions on the same rectangular coordinate system. $$ x^{2}+2 x y+y^{2}+2 x-4 y=5 $$

Step-by-Step Solution

Answer

The equation describes a parabola, identified by the discriminant.

Step 1: Rewrite the Equation

The given equation is \[ x^2 + 2xy + y^2 + 2x - 4y = 5 \]To rewrite this, move all terms to one side and set it equal to zero:\[ x^2 + 2xy + y^2 + 2x - 4y - 5 = 0 \]

Step 2: Identify Conic Using the Discriminant

The general second-degree equation is \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]For our equation, we have:- \( A = 1 \)- \( B = 2 \)- \( C = 1 \)- \( D = 2 \)- \( E = -4 \)- \( F = -5 \)The discriminant is given by \( D = B^2 - 4AC \):\[ D = 2^2 - 4 \cdot 1 \cdot 1 = 4 - 4 = 0 \]When the discriminant \( D = 0 \), the conic is a parabola.

Step 3: Rewrite the Equation in Standard Form

The equation \[ x^2 + 2xy + y^2 + 2x - 4y = 5 \]can be rewritten in a more standard form for a parabola. To isolate terms involving only \( x \) or \( y \), complete the square where applicable. For now, observe that we need implicit functions out of the equation. Let's try rewriting by assuming possible functions based on variable groupings.

Step 4: Find Implicit Functions

Given that the equation describes a conic with mixed \( x \) and \( y \) terms, finding explicit functions involves solving for one variable in terms of the other. For example, solve for \( y \):Rearrange the terms into a more explicit form and assume function separation:\[ y^2 + 2xy - 4y = 5 - x^2 - 2x \]Try solutions of form \( y = f(x) \) using techniques like isolating \( y \) from mixed terms.

Step 5: Determine Domain of Each Function

For each of the implicit functions derived, calculate the domain by ensuring that for all values of \( x \) within domain, \( y(x) \) yields real numbers:Given the implicit nature, determine each function numerically or graphically using technology, ensuring consistent, real \( y \)-values, and discuss constraints directly stemming from the functions.

Step 6: Graph the Functions

Use a graphing utility to plot the obtained functions on a coordinate system. Since the conic is a parabola, these functions will likely map out the branches of this conic.

Key Concepts

Discriminant

The discriminant is a powerful tool when working with conic sections. It helps determine the type of conic section represented by a quadratic equation. Imagine you have a general second-degree equation of the form

• \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]

To reveal the conic's nature, we calculate the discriminant using:

• \[ D = B^2 - 4AC \]

The result tells us what type of curve we're dealing with:

• If \( D > 0 \), it's a hyperbola.
• If \( D = 0 \), the equation represents a parabola.
• If \( D < 0 \), it's an ellipse. (And when \( A = C \) and \( B = 0 \), it's a circle.)

For the equation \( x^2 + 2xy + y^2 + 2x - 4y = 5 \), we found \( D = 0 \), confirming that it depicts a parabola. Calculating the discriminant is crucial to understand the overall structure and graphing behavior of the conic.

Implicit Functions

Implicit functions are useful when direct algebraic solutions for variables are not easy to obtain. Instead of expressing one variable explicitly in terms of another, implicit functions allow them to coexist in an equation naturally. Consider our given equation,

• \[ x^2 + 2xy + y^2 + 2x - 4y = 5 \]

This expression involves both \( x \) and \( y \) mixed together without isolating \( y \) or \( x \). Implicit function theory suggests breaking the equation into manageable parts. For instance,

• By assuming forms like \( y = f(x) \), which might involve rearranging and factoring parts of the equation
• Numerically solving parts to gather insight

However, these methods can be complex and sometimes require technology to solve. That's because implicit functions might not always display clear domains without further exploration.Despite the hurdle, utilizing implicit functions is invaluable for equations where variables are intertwined, as this method respects their original relationship.

Graphing Utility

A graphing utility is a digital tool designed to aid in visualizing mathematical equations and functions. When dealing with conic sections, such as parabolas, graphing utilities become indispensable for exploring solutions, especially for complex or implicit functions.Let's say we have the earlier equation

• \[ x^2 + 2xy + y^2 + 2x - 4y = 5 \]

Since this is a parabola, plotting it through a graphing utility can simplify the interpretation of its lobes and curves. Here's how it helps:

• Visualizing interactions between \( x \) and \( y \), clearly displaying the conic shape
• Finding and confirming values that make the equation true by adjusting inputs
• Providing an accurate depiction of the domain and range, as it automatically adjusts the axes

Interactive graphing utilities allow the exploration of different parts of the equation smoothly, breaking down complex relationships into understandable plots. These tools not only affirm algebraic manipulation but enhance overall comprehension through visual engagement.