교과서 질문
Identify each equation without completing the square.
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Ch. 7 - Conic Sections
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
8장, 문제 55
In Exercises 51–56, graph each relation. Use the relation's graph to determine its domain and range.
검증된 단계별 안내1
Recognize that the given equation \(\frac{y^2}{16} - \frac{x^2}{9} = 1\) represents a hyperbola centered at the origin. This is because it matches the standard form of a vertical hyperbola: \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\) where \(a^2 = 16\) and \(b^2 = 9\).
Identify the values of \(a\) and \(b\) by taking square roots: \(a = 4\) and \(b = 3\). These values help determine the shape and orientation of the hyperbola.
To graph the hyperbola, plot the vertices on the \(y\)-axis at \((0, \pm a)\), which are \((0, \pm 4)\). These points are where the hyperbola intersects the \(y\)-axis.
Determine the domain by solving for \(x\) in terms of \(y\). Rearrange the equation to isolate \(x^2\): \(\frac{y^2}{16} - 1 = \frac{x^2}{9}\). Then multiply both sides by 9 to get \(x^2 = 9\left(\frac{y^2}{16} - 1\right)\). For \(x^2\) to be non-negative (real \(x\) values), the expression inside the parentheses must be greater than or equal to zero. Use this to find the range of \(y\) values that produce real \(x\) values, which will help determine the domain of the relation.
Similarly, determine the range by considering the possible \(y\) values. Since the hyperbola opens vertically, \(y\) can take any real value such that the expression under the square root for \(x\) is valid. Use the inequality from the previous step to find the exact range of \(y\) values.
주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Hyperbola Equation and Standard Form
A hyperbola is a conic section defined by an equation where the difference of squared terms equals a constant, such as \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \). This form indicates a vertical transverse axis, with vertices at \( (0, \pm a) \). Understanding this standard form helps in identifying the shape and orientation of the graph.
추천 영상:
5:50
Asymptotes of Hyperbolas
Domain and Range of Relations
The domain of a relation is the set of all possible x-values, while the range is the set of all possible y-values. For hyperbolas, these sets depend on the equation's restrictions and asymptotes. Determining domain and range from the graph involves analyzing where the curve exists along the x- and y-axes.
추천 영상:
4:22Domain & Range of Transformed Functions
Graphing Conic Sections
Graphing conic sections like hyperbolas requires plotting key points such as vertices and asymptotes, which guide the curve's shape. For the given equation, asymptotes are lines that the hyperbola approaches but never touches, found by setting the equation equal to zero. Accurate graphing aids in visualizing the relation and extracting domain and range.
추천 영상:
3:08Geometries from Conic Sections
관련 실천
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교과서 질문
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Identify each equation without completing the square.
557
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