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Chapter 10: Problem 27

Determine the vertex, focus, and directrix for each parabola. $$y=\frac{1}{4}(x-3)^{2}$$

Short Answer

Expert verified
The vertex is at (3, 0), the focus is at (3, 1), and the directrix is the line y = -1.

Step by step solution

01

Write the equation in standard form

The given equation is already in vertex form, which is similar to the standard form for a parabola opening upwards or downwards:\[ y = a(x-h)^2 + k \]Here, the equation is \( y = \frac{1}{4}(x-3)^2 \), so we identify \( h = 3 \), \( k = 0 \), and \( a = \frac{1}{4} \).
02

Determine the vertex

The vertex form of the parabola \( y = a(x-h)^2 + k \) gives the vertex directly as \( (h, k) \).Thus, the vertex is at \( (3, 0) \).
03

Find the value of p

The parameter \( p \) is related to \( a \) by the equation \( a = \frac{1}{4p} \). Solving for \( p \) with \( a = \frac{1}{4} \):\[ \frac{1}{4} = \frac{1}{4p} \]\[ p = 1 \]
04

Determine the focus

The focus of a parabola \( y = a(x-h)^2 + k \) is given by \( (h, k+p) \). Here, \( h = 3 \), \( k = 0 \), and \( p = 1 \), so the focus is at:\[ (3, 0+1) = (3, 1) \]
05

Determine the directrix

The directrix of a parabola \( y = a(x-h)^2 + k \) is given by the line \( y = k-p \). With \( k = 0 \) and \( p = 1 \), the directrix is:\[ y = 0-1 = -1 \]

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Key Concepts

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

Understanding the Vertex Form of a Parabola
The vertex form of a parabola is a very useful equation. It allows us to easily identify the vertex of the parabola. A parabola in vertex form is written as: \[ y = a(x-h)^2 + k \]In this equation,
  • \(a\) determines the width and direction of the parabola.
  • \(h\) and \(k\) are the coordinates of the vertex.
For example, consider the equation you provided:\[ y = \frac{1}{4}(x-3)^2 \]Here, by comparing the given equation with the standard vertex form, we can see that:
  • \(h = 3\)
  • \(k = 0\)
  • \(a = \frac{1}{4}\)
Thus, the vertex of this parabola is at (3, 0). This is a significant first step as knowing the vertex gives us crucial information about the shape and position of the parabola.
Determining the Focus of the Parabola
The focus of a parabola is another important feature. It is a point from which distances are measured to define the parabola.To find the focus, we need to determine the parameter \(p\). The relation between \(a\) and \(p\) is:\[ a = \frac{1}{4p} \]Given \(a = \frac{1}{4}\), we solve for \(p\):\[ \frac{1}{4} = \frac{1}{4p} \]\[ p = 1 \]With \(h = 3\), \(k = 0\), and \(p = 1\), we can find the focus using the formula for the coordinates of the focus in the vertex form equation:\[ (h, k + p) \]So, the focus is at (3, 1). This point lies inside the parabola and helps in defining its reflective properties.
Finding the Directrix of the Parabola
The directrix is a line that is used along with the focus to define a parabola. Every point on the parabola is equidistant to the focus and the directrix.For the equation \[ y = a(x-h)^2 + k \]the directrix is given by the line:\[ y = k - p \]Substituting the values we have:\[ k = 0 \]\[ p = 1 \]The directrix can be found as:\[ y = 0 - 1 = -1 \]Thus, the directrix is the horizontal line \( y = -1 \). This line, along with the focus, provides a complete description of the geometric properties of the parabola.

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Most popular questions from this chapter

Find the vertex, focus, and directrix of each parabola without completing the square, and determine whether the parabola opens upward or downward. $$y=-x^{2}+4 x+3$$

Write each of the following equations in one of the forms: \(y=a(x-h)^{2}+k, \quad x=a(y-h)^{2}+k\) \(\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1,\) or \((x-h)^{2}+(y-k)^{2}=r^{2}\). Then identify each equation as the equation of a parabola, an ellipse, or a circle. $$\frac{x^{2}}{4}+\frac{y^{2}}{4}=1$$

Write the equation for each circle described. Center \((0,0)\) and passing through \((4,5)\)

Determine the center and radius of each circle and sketch its graph. $$x^{2}+y^{2}=25$$

Write the equation for each circle described. Center \((0,0)\) and passing through \((-3,-4)\)
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