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Chapter 13: Problem 85

Find the coordinates of the vertex and the direction in which each parabola opens. A. \(y=8(x-3)^{2}+6\) B. \(x=8(y-3)^{2}+6\)

Short Answer

Expert verified
A: Vertex is (3, 6), opens upwards; B: Vertex is (6, 3), opens to the right.

Step by step solution

01

Identify the Vertex Form

The vertex form of a parabola equation can help us quickly identify the vertex coordinates. For the equation \( y = a(x-h)^2 + k \), the vertex is at \((h, k)\). Likewise, for \( x = a(y-k)^2 + h \), the vertex is at \((h, k)\).
02

Determine the Vertex of Equation A

The given equation is \( y = 8(x-3)^2 + 6 \). This matches the vertex form \( y = a(x-h)^2 + k \), where \( h = 3 \) and \( k = 6 \). Therefore, the vertex is \((3, 6)\).
03

Determine the Direction of Opening for Equation A

Since the equation is in the form of \( y = a(x-h)^2 + k \) and the coefficient \( a = 8 \) is positive, the parabola opens upwards.
04

Determine the Vertex of Equation B

The equation is \( x = 8(y-3)^2 + 6 \). This fits the form \( x = a(y-k)^2 + h \), where \( h = 6 \) and \( k = 3 \). Therefore, the vertex is \((6, 3)\).
05

Determine the Direction of Opening for Equation B

In this case, the form \( x = a(y-k)^2 + h \) is used, where the coefficient \( a = 8 \) is positive, meaning the parabola opens to the right.

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Key Concepts

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

Vertex Form
The vertex form of a parabola is an elegant way to express the equation of a parabola because it allows us to easily identify important features of the parabola. There are two main types of parabolic equations that we encounter:
  • For the equation of the form \(y = a(x-h)^2 + k\), the vertex is located at the point \((h, k)\). This type of equation describes a parabola that opens either upwards or downwards, depending on the value of \(a\).
  • For the equation of the form \(x = a(y-k)^2 + h\), the vertex is at the point \((h, k)\) as well, but this describes a parabola that opens either to the right or left, again depending on \(a\).
The vertex is a crucial part of a parabola because it represents the maximum or minimum point of the parabola, known as the "turning point." This point is where the parabola changes its direction.
Direction of Opening
Understanding the direction in which a parabola opens is fundamental to grasping its geometry. The direction of opening is determined primarily by the coefficient \(a\) in the vertex form. Here鈥檚 how it works:
  • For a vertical parabola given by \(y = a(x-h)^2 + k\):
    • If \(a > 0\), the parabola opens upwards.
    • If \(a < 0\), the parabola opens downwards.
  • For a horizontal parabola expressed as \(x = a(y-k)^2 + h\):
    • If \(a > 0\), it opens to the right.
    • If \(a < 0\), it opens to the left.
The sign of \(a\) is crucial. It doesn鈥檛 just determine the opening direction, but it also hints at other properties, such as whether the vertex is a maximum or minimum point in vertical parabolas.
Parabolic Equations
Parabolic equations describe curves known as parabolas, which are U-shaped or directionally similar curves. They feature prominently in various mathematical applications and scenarios. Parabolas can be aligned either vertically or horizontally:
  • In vertical parabolas, the equation is written as \(y = a(x-h)^2 + k\) and the graph is symmetric around the vertical axis passing through the vertex.
  • In horizontal parabolas, the form \(x = a(y-k)^2 + h\) means the graph is symmetric about a horizontal axis through the vertex.
These equations not only simplify identifying properties like the vertex and opening direction but are also pivotal in physics for modeling projectile motion and other phenomena. Parabolic equations are straightforward yet powerful representations with wide real-world applications.

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Most popular questions from this chapter

Solve each system of equations by elimination for real values of x and y. $$ \left\\{\begin{array}{l} x^{2}+y^{2}=20 \\ x^{2}-y^{2}=-12 \end{array}\right. $$

Write each equation in standard form, if it is not alreacty so, and graph it. If the graph is a circle, give the coordinates of its center and its radius. If the graph is a parabola, give the coordinates of its vertex. $$ y=-4(x+5)^{2}+5 $$

Solve each system of equations for real values of x and y. $$ \left\\{\begin{array}{l} x-y=-1 \\ y^{2}-4 x=0 \end{array}\right. $$

Solve each system of equations by substitution for real values of x and y. $$ \left\\{\begin{array}{l} x^{2}+y^{2}=10 \\ y=3 x^{2} \end{array}\right. $$

Write the equation of a circle with a diameter whose endpoints are at \((-5,4)\) and \((7,-3)\)
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