Skip to main content
Ch. 7 - Conic Sections
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 7 - Conic SectionsProblema 57
Capitolo 8, Problema 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

Guida verificata passo dopo passo
1
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.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
8m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

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.
Video consigliato:
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.
Video consigliato:
4:22
Domain & 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.
Video consigliato:
5:20
Relations and Functions
Pratica correlata
Domanda del libro di testo

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?

x=−4(y−1)2+3x = - 4(y - 1)^2 + 3


736
views
Domanda del libro di testo

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

827
views
Domanda del libro di testo

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

779
views
Domanda del libro di testo

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
views
Domanda del libro di testo

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?

y=−x2+4x−3y=-x^2+4x-3


732
views
Domanda del libro di testo

Identify each equation without completing the square.

y2+8x+6y+25=0y^2+8x+6y+25=0

557
views