/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 30 Classify the following as the eq... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 30

Classify the following as the equation of a circle, an ellipse, a parabola, or a hyperbola. \(x+3 y=2 y^{2}-1\)

Short Answer

Expert verified
Parabola

Step by step solution

01

Rearrange the equation

Start by rearranging the given equation to group the terms with the same variables together. Given equation: \(x + 3y = 2y^{2} - 1\)Move all terms to one side of the equation to set it to 0:\(x + 3y - 2y^{2} + 1 = 0\)
02

Identify the degree of each term

Identify the degree of each term in the equation:- \(x\) is linear (degree 1)- \(3y\) is linear (degree 1)- \(-2y^{2}\) is quadratic (degree 2)- \(1\) is constant (degree 0)
03

Analyze the highest degree term

Notice that the highest degree term is \(-2y^{2}\), which is quadratic. There is no \(x^{2}\) term. Since the highest degree is 2 and we have both linear and quadratic terms, this suggests a parabola.
04

Conclusion

Based on the rearranged equation \(x + 3y - 2y^{2} + 1 = 0\) and the identification of terms, we can classify this as the equation of a parabola.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Equation Classification
Understanding how to classify different equations is an essential skill in algebra and geometry. Different types of equations represent various geometric shapes like circles, ellipses, parabolas, and hyperbolas. In this context, the classification is based on the terms present in the equation and their powers. For instance:
  • Equations with squared terms for both variables generally represent ellipses or hyperbolas, depending on the signs and coefficients.
  • If there鈥檚 only one squared variable term, as in the given problem, the equation is likely to represent a parabola.
  • Equations without squared terms but with both variables usually represent straight lines.
This classification helps understand the geometric representation and behavior of the equation. By examining the equation, we can identify the type of graph it will produce, making the problem more manageable.
Quadratic Terms
Quadratic terms are an integral part of polynomial expressions and play a crucial role in classifying conic sections. A quadratic term involves squaring a variable, and it has the form \(ax^2 \) or \(ay^2\). In our example, \(-2y^2\) is the quadratic term and is the highest degree term since its power is 2.
  • Quadratic terms are always parabolic in nature, contributing to the 'curving' of the graph.
  • If a quadratic term involves only one variable, it typically results in a parabolic shape on the graph.
  • The coefficient before the squared term (like \(-2\) in \(-2y^2\)) affects the 'opening' and 'direction' of the parabola.
Two quadratic terms, such as \(x^2\) and \(y^2\),exist in equations of ellipses, circles, or hyperbolas. Identifying and understanding these terms ensure accurate graph classification and interpretation.
Linear Terms
Linear terms involve variables raised to the first power, making them crucial for understanding the overall shape of the equation. They appear as \(ax\) or \(by\), where the exponent of the variable is 1. In the example given, both \(x\) and \(3y\) are linear terms.
  • Linear terms contribute to the 'direction' and position of the graph on the coordinate plane.
  • In combination with quadratic terms, linear terms help define the specific type of conic section. For instance, having \(x\) along with \(y^2\) often indicates a parabola.
  • Linear terms can shift a graph left, right, up, or down, affecting the 'vertex' of a parabola or center of a circle or ellipse.
Understanding linear terms alongside quadratic terms allows for precise equation classification and helps in sketching the graph correctly.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find an equation of a circle satisfying the given conditions. Center \((-3,5)\) with a circumference of \(8 \pi\) units

Match the equation with the center or vertex of its graph, listed in the column on the right. a) Vertex: \((-2,5)\) b) Vertex: \((5,-2)\) c) Vertex: \((2,-5)\) d) Vertex: \((-5,2)\) e) Center: \((-2,5)\) f) Center: \((2,-5)\) g) Center: \((5,-2)\) h) Center: \((-5,2)\) $$y=(x-5)^{2}-2$$

Firefighting. The size and shape of certain forest fires can be approximated as the union of two "halfellipses." For the blaze modeled below, the equation of the smaller ellipse - the part of the fire moving into the wind- is $$ \frac{x^{2}}{40,000}+\frac{y^{2}}{10,000}=1 $$ The equation of the other ellipse - the part moving with the wind- is $$ \frac{x^{2}}{250,000}+\frac{y^{2}}{10,000}=1 $$ Determine the width and the length of the fire. Source for figure: "Predicting Wind-Driven Wild Land Fire Size and Shape," Hal E. Anderson. Rescarch Paper INT-305. U.S. Department of Agriculture, Forest Service, February 1983

Four squares with sides 5 in. long are cut from the corners of a rectangular metal sheet that has an area of 340 in \(^{2} .\) The edges are bent up to form an open box with a volume of 350 in \(^{3} .\) Find the dimensions of the box.

Astronomy. The earth's orbit around the sun is an ellipse with \(a \approx 149.7\) million \(\mathrm{km} .\) The sun, located at one focus of the ellipse, is approximately 2.4 million km from the center of the ellipse. What is the maximum distance of the earth from the sun? (IMAGE CANNOT COPY)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.