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Chapter 7: Problem 49

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. Is the relation a function? $$y^{2}+6 y-x+5=0$$

Short Answer

Expert verified
The vertex of the parabola is \((-3/2, -21/4)\). The parabola opens to the right. The domain of the parabola is \([-21/4, +\infty)\) and the range is \((-\infty, +\infty)\). The relation given by the equation is not a function.

Step by step solution

01

Rearrange the equation

Rearrange the equation as follows: \(x=y^{2}+6y-5\), which can be recognized as the standard form \((y-k)^{2}=4p(x-h)\) of the equation of a parabola oriented in a horizontal direction.
02

Complete the square

Complete the square on \(y\). To do so, add \(9/4\) to both sides of the equation to get \(x+5+(3/2)^2=y^{2}+6y+(3/2)^2\). After simplifying, this reads as \(x+41/4=(y+3/2)^2\). Finally, convert back to standard form to get \((y+3/2)^2=x+41/4 - 5\), which simplifies to \((y+3/2)^2=x+21/4\).
03

Identify Vertex and Direction

The equation, \((y+3/2)^2=x+21/4\), shows a parabola with vertex at \((-3/2, -21/4)\) opening to the right since x is positively related to \(y^2\).
04

Determine Domain and Range

For a parabola opening to the right, the domain is \([h, +\infty)\) and the range is \((-\infty, +\infty)\). For this case, the domain is \([-21/4, +\infty)\) and the range is \((-\infty, +\infty)\).
05

Determine if the relation is a Function

Since every x-value corresponds with multiple y-values, this relation is not a function.

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Most popular questions from this chapter

Convert each equation to standard form by completing the square on \(x\) or \(y .\) Then find the vertex, focus, and directrix of the parabola. Finally, graph the parabola. $$y^{2}-2 y-8 x+1=0$$

Find the standard form of the equation of each parabola satisfying the given conditions. Focus: \((7,0) ;\) Directrix: \(x=-7\)

Find the standard form of the equation of each parabola satisfying the given conditions. Focus: \((-10,0) ;\) Directrix: \(x=10\)

If you are given the standard form of the equation of a parabola with vertex at the origin, explain how to determine if the parabola opens to the right, left, upward, or downward.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I noticed that depending on the values for \(A\) and \(B\), assuming that they are both not zero, the graph of \(A x^{2}+B y^{2}=C\) can represent any of the conic sections other than a parabola.
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