Domanda del libro di testo
Find the vertex, focus, and directrix of the parabola with the given equation. Then graph the parabola. x^2 - 4x - 2y = 0
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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
Capitolo 8, Problema 41
Find the standard form of the equation of the parabola satisfying the given conditions. Focus: (12,0); Directrix: x=-12
Guida verificata passo dopo passo1
Identify the general form of a parabola with a horizontal axis of symmetry. The standard form is: , where is the vertex, and is the distance from the vertex to the focus or the directrix.
Determine the vertex of the parabola. The vertex lies midway between the focus and the directrix. Since the focus is at and the directrix is , calculate the midpoint between these two points to find the vertex.
Calculate the value of , which is the distance from the vertex to the focus. Use the formula .
Substitute the vertex coordinates and the value of into the standard form equation .
Simplify the equation to express it in standard form, ensuring all terms are properly arranged and simplified.
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Standard Form of a Parabola
The standard form of a parabola that opens horizontally is given by the equation (y - k)² = 4p(x - h), where (h, k) is the vertex and p is the distance from the vertex to the focus. This form allows for easy identification of the parabola's orientation and key features, such as the focus and directrix.
Video consigliato:
5:33
Parabolas as Conic Sections
Focus and Directrix
In the context of parabolas, the focus is a fixed point from which distances to points on the parabola are measured, while the directrix is a line that is perpendicular to the axis of symmetry. The parabola is defined as the set of points equidistant from the focus and the directrix, which is crucial for determining its equation.
Video consigliato:
5:33Parabolas as Conic Sections
Vertex of the Parabola
The vertex of a parabola is the point where it changes direction and is located midway between the focus and the directrix. For a parabola that opens horizontally, the vertex can be calculated as the midpoint of the focus and the directrix, which is essential for writing the standard form of the parabola's equation.
Video consigliato:
5:28Horizontal Parabolas
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