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Chapter 13: Problem 30

Identify whether each equation, when graphed, will be a parabola, circle, ellipse, or hyperbola. Sketch the graph of each equation. \(x=y^{2}+4 y-1\)

Short Answer

Expert verified
The given equation represents a horizontally-opening parabola.

Step by step solution

01

Identify the equation form

Given the equation \(x = y^{2} + 4y - 1\), this needs to be analyzed to identify its form. It resembles the form of \(x = a(y - k)^2 + h\), which is a parabola that opens horizontally.
02

Recognize coefficients and complete the square

In the equation \(x = y^2 + 4y - 1\), we can complete the square for \(y\). This helps us rewrite it in vertex form. Start by dealing with \(y^2 + 4y\): \((y^2 + 4y) = (y+2)^2 - 4\).
03

Rewrite the equation in vertex form

Substitute \((y+2)^2 - 4\) back into the equation: \(x = (y+2)^2 - 4 - 1\). Simplifying gives \(x = (y+2)^2 - 5\). This confirms it's a parabola.
04

Determine the direction and vertex

Since the equation is \(x = (y+2)^2 - 5\), it's a horizontally-opening parabola. The vertex is at \((-5, -2)\).
05

Sketch the graph

Plot the vertex at \((-5, -2)\). Since it opens horizontally, draw the parabolic shape with the axis of symmetry parallel to the y-axis. It opens in the positive x-direction.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Completing the Square
Completing the square is a useful technique in algebra that helps transform a quadratic equation into a form that is easier to analyze, especially when identifying specific features like the vertex of a parabola. When given the equation \(x = y^2 + 4y - 1\), you first focus on the quadratic terms involving \(y\). Completing the square involves:
  • Focusing on the expression \(y^2 + 4y\).
  • Taking half of the coefficient of \(y\), which in this case is 4, resulting in 2, and then squaring it to get 4.
  • Rewriting \(y^2 + 4y\) as \((y + 2)^2 - 4\).
By inserting this into the original equation, we are left with \(x = (y + 2)^2 - 5\). This transformation was achieved through completing the square, making it easier to convert the equation into the vertex form of a parabola.
Vertex Form
The vertex form of a parabolic equation is a reconceptualization that makes it simple to analyze and graph parabolas. It is generally written as \( x = a(y - k)^2 + h \) for parabolas opening horizontally, where \( (h, k) \) is the vertex of the parabola.When we rewrite \( x = y^2 + 4y - 1 \) by completing the square, we get \( x = (y + 2)^2 - 5 \). This equation is now in the vertex form.The benefits of having a parabola in vertex form include:
  • Identifying the vertex directly; here it is \((-5, -2)\).
  • Understanding the direction of the opening: this parabola opens horizontally because the \(y\)-terms are squared.
  • Visualizing the parabola's graph becomes easier since the vertex is a key reference point.
Vertex form simplifies the sketching process, providing clear insights into the parabola's attributes.
Conic Sections
Conic sections are the curves obtained by intersecting a plane with a double-napped cone. The shape and nature of the conic depend on the angle of the intersection.In mathematics, four primary types of conic sections can occur:
  • Circle: A conic section with all points equidistant from the center.
  • Ellipse: An elongated circle formed when the plane cuts the cone at an angle to its base.
  • Parabola: Formed when the plane is parallel to the side of the cone. This is the type relevant to our equation \(x = y^2 + 4y - 1\).
  • Hyperbola: Created when the plane intersects both halves of the cone.
Parabolas, like the one in our example, are unique because they have a single axis of symmetry and their curves can open up, down, left, or right. Recognizing the type of conic section helps in understanding its properties and in sketching it accurately. In the given exercise, the equation describes a horizontally opening parabola, a classic example of a conic section.

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Most popular questions from this chapter

Sketch the graph of each equation. If the graph is a parabola, find irs vertex. If the graph is a circle, find its center and radius. $$x=-2 y^{2}-4 y$$

Solve. The sum of the squares of two numbers is \(130 .\) The difference of the squares of the two numbers is \(32 .\) Find the two numbers.

The graph of each equation is a parabola. Find the vertex of the parabola and sketch its graph. See Examples I through 4. $$ x=(y-2)^{2}+3 $$

The graph of each equation is a parabola. Find the vertex of the parabola and sketch its graph. See Examples I through 4. $$ x=3 y^{2} $$

Sketch the graph \(\frac{(x+5)^{2}}{16}-\frac{(y+2)^{2}}{25}=1\)
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