교과서 질문
In Exercises 57–62, use the vertex and the direction in which the parabola opens to determine the relation's domain and range. Is the relation a function?
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Ch. 7 - Conic Sections
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- 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 57
Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. Is the relation a function? y2 + 6y - x + 5 = 0
검증된 단계별 안내1
Rewrite the given equation \(y^2 + 6y - x + 5 = 0\) to express \(x\) in terms of \(y\). Start by isolating \(x\): \(x = y^2 + 6y + 5\).
Complete the square for the \(y\)-terms to rewrite the equation in vertex form. Take \(y^2 + 6y\) and complete the square: \(y^2 + 6y = (y + 3)^2 - 9\).
Substitute the completed square back into the expression for \(x\): \(x = (y + 3)^2 - 9 + 5 = (y + 3)^2 - 4\).
Identify the vertex of the parabola from the equation \(x = (y + 3)^2 - 4\). The vertex is at \((-4, -3)\), and since \(x\) is expressed as a square of \((y + 3)\), the parabola opens horizontally (to the right).
Determine the domain and range: Since the parabola opens horizontally, the domain is all \(x\) values greater than or equal to the vertex's \(x\)-coordinate, and the range is all real \(y\) values. Finally, check if for each \(x\) there is only one \(y\) value to decide if the relation is a function.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Rearranging and Identifying the Parabola's Form
To analyze the given relation, rewrite the equation to isolate x or y. Completing the square for the y-terms helps express the equation in vertex form, revealing the parabola's vertex and orientation. This step is crucial for understanding the graph's shape and properties.
추천 영상:
5:28
Horizontal Parabolas
Domain and Range of Parabolas
The domain is the set of all possible x-values, and the range is the set of all possible y-values of the relation. For parabolas, the vertex and the direction it opens (left, right, up, or down) determine these intervals. Understanding how the parabola extends helps identify domain and range accurately.
추천 영상:
4:22Domain & Range of Transformed Functions
Determining if the Relation is a Function
A relation is a function if each input (x-value) corresponds to exactly one output (y-value). For parabolas that open sideways (like this one), some x-values may have two y-values, violating the function definition. Using the vertical line test or analyzing the equation helps decide if the relation is a function.
추천 영상:
5:20Relations and Functions
관련 실천
교과서 질문
Convert each equation to standard form by completing the square on x and y. Then graph the ellipse and give the location of its foci. 25x²+4y² – 150x + 32y + 189 = 0
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교과서 질문
Convert each equation to standard form by completing the square on x and y. Then graph the ellipse and give the location of its foci. 4x² + y²+ 16x - 6y - 39 = 0
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교과서 질문
Convert each equation to standard form by completing the square on x and y. Then graph the ellipse and give the location of its foci. 36x2 +9y2 - 216x = 0
827
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교과서 질문
In Exercises 57–62, use the vertex and the direction in which the parabola opens to determine the relation's domain and range. Is the relation a function?
732
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교과서 질문
Identify each equation without completing the square.
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