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Chapter 9: Problem 51

Use a graphing utility to graph each equation. $$7 x^{2}+6 x y+2.5 y^{2}-14 x+4 y+9=0$$

Short Answer

Expert verified
Without simplification to a standard form of an ellipse, parabola or a hyperbola, the exact conic section represented by the given equation isn't immediately clear. By plotting it with a graphing tool, the shape revealed will provide the answer.

Step by step solution

01

Rewriting the Equation

The given equation is a quadratic form. To facilitate graphing, we can rewrite this conic equation in its standard form. But since a simplification like this isn't easily possible in this case, we will use the equation as is to plot it.
02

Input the Equation into the Graphing Utility

Following step 1, we input the provided equation into a graphing utility exactly as it is. The graphing utility is a tool that allows the plotting of complex mathematical formulas. There are several free online tools that can create graphic interpretations of complex equations.
03

Interpret the Graph

After inputting the equation into the graphing utility, one can clearly see the resultant conic section. From the graph, it should be clear what kind of shape the equation represents, whether it is an ellipse, a parabola, or a hyperbola.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Equation
To begin understanding conic sections, we must first discuss quadratic equations. A quadratic equation is any equation that can be rearranged in the form:
  • \(ax^2 + bxy + cy^2 + dx + ey + f = 0\)
This equation is useful in graphing various conic sections, including ellipses, parabolas, and hyperbolas. The presence of the term \(bxy\) suggests a rotation of axes, which can make these equations trickier to interpret by sight alone.
Quadratic equations are foundational as they allow for the geometric representation of shapes. Understanding how changes in coefficients affect the overall graph is key to mastering conics.
Graphing Utility
Graphing utilities are powerful tools that enable us to visualize complex equations without manually plotting every point. By inputting equations directly into these tools, you can produce accurate graphs efficiently.
These utilities can handle complex expressions and offer various features such as zooming and toggling grid views to better analyze graphs.
  • Diversified tools: Both online and offline options are available.
  • Precision: Decimal and fractional value settings ensure accuracy.
  • Interactive: Allows manipulation to see immediate changes.
Using these tools not only saves time but also aids in developing a deeper understanding of geometric relationships in equations.
Ellipse
An ellipse is a type of conic section that resembles an elongated circle. It is defined by its unique shape, where all points on the ellipse are the collective focus of two points known as foci. The standard form of an ellipse can be expressed as:
  • \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\)
Here, \((h, k)\) represents the center, and \(a\) and \(b\) are the semi-major and semi-minor axes, respectively.
Ellipses have a variety of applications, particularly in physics, where planetary orbits often approximate elliptical paths. Recognizing the shape of an ellipse in a graphing utility can help identify the involved relationship within the equation and further study its characteristics.
Parabola
A parabola is another common conic section with a distinctive U-shaped curve. It is typically expressed in the form:
  • \(y = ax^2 + bx + c\)
or its vertex form:
  • \(y - k = a(x-h)^2\)
where \((h, k)\) is the vertex of the parabola. Every parabola has a directrix and a focus; these properties allow the parabola to have reflective qualities. Depending on the values of \(a, b,\) and \(c\), the parabola can open upward, downward, or sideways.
Parabolas have practical applications in areas such as physics and engineering, especially in the design of satellite dishes and car headlights, which rely on the reflective properties of parabolas.
Hyperbola
Unlike ellipses and parabolas, hyperbolas consist of two separate open curves or branches. These curves are defined by their asymptotes, which provide the direction of the hyperbola's opening. The standard form of a hyperbola is:
  • \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\)
or the reverse, which is:
  • \(\frac{(y-k)^2}{b^2} - \frac{(x-h)^2}{a^2} = 1\)
Foci for hyperbolas are outside the curve, helping in identifying the branches of the graph.
Hyperbolas have important applications in different fields, like acoustics and navigation systems. Recognizing a graph as a hyperbola helps interpret the relationships described by the equation you're examining.

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Most popular questions from this chapter

What does it mean to eliminate the parameter? What useful information can be obtained by doing this?

Solve the system: $$\left\\{\begin{aligned} y &=x^{2}-7 \\ x^{2}+y^{2} &=13 \end{aligned}\right.$$

In Exercises \(41-43\), eliminate the parameter. Write the resulting equation in standard form. A circle: \(x=h+r \cos t, y=k+r \sin t\)

In Exercises \(63-68\), sketch the function represented by the given parametric equations. Then use the graph to determine each of the following: a. intervals, if any, on which the function is increasing and intervals, if any, on which the function is decreasing. b. the number, if any, at which the function has a maximum and this maximum value, or the number, if any, at which the function has a minimum and this minimum value. $$x=\frac{t}{2}, y=2 t^{2}-8 t+3$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. In order to graph an ellipse whose equation contained an \(x y\) -term, I used a rotated coordinate system that placed the ellipse's center at the origin.
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