/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 Draw the curves \(3-14\) by hand... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 6

Draw the curves \(3-14\) by hand or calculator or computer. Locate the vertices and foci. $$ 4 x=y^{2} $$

Short Answer

Expert verified
The vertex is at \((0, 0)\) and the focus is at \((1, 0)\); the parabola opens to the right.

Step by step solution

01

Identify the Type of Curve

The given equation is of the form \( 4x = y^2 \), which can be rewritten as \( x = \frac{1}{4}y^2 \). This equation represents a parabola, which opens sideways (to the right) because the \( y^2 \) term is isolated on one side.
02

Rewrite in Standard Form

The standard form for a sideways parabola is \( (y - k)^2 = 4p(x - h) \). By comparing this with \( y^2 = 4x \), we can see that the equation is already in standard form with \( h = 0 \) and \( k = 0 \), meaning the vertex is at the origin \((0,0)\). The parameter \( p \) is \( 1 \), since \( 4p = 4 \).
03

Determine the Vertex

The vertex of the parabola is at the point \((h, k)\). Since the numbers \(h = 0\) and \(k = 0\) were found in the standard form conversion, the vertex of this parabola is at \((0, 0)\).
04

Locate the Focus

For the parabola \((y - k)^2 = 4p(x - h)\), the focus is located at \((h + p, k)\). Since \(p = 1\), \(h = 0\), and \(k = 0\), the focus is at \((0 + 1, 0)\) or \((1, 0)\).
05

Draw the Parabola

The parabola opens to the right. Begin at the vertex \((0, 0)\), plot the focus at \((1, 0)\), and draw a symmetric curve passing through the focus, opening to the right. Use a graphing calculator or plot points to accurately sketch the curve.

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Key Concepts

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

Standard Form of Parabola
The standard form of a parabola can help you determine the parabola's shape, position, and direction. For a parabola that opens either left or right, the equation is \[ (y - k)^2 = 4p(x - h) \] Here,
  • \( (h, k) \) is the vertex of the parabola.
  • \( p \) determines the distance from the vertex to the focus and the directrix.
When you identify that given equation \( 4x = y^2 \) corresponds to \( x = \frac{1}{4}y^2 \), it matches the form \( (y - 0)^2 = 4(\frac{1}{4})(x - 0) \). Hence, it is clear that the parabola opens towards the right.
Vertex of Parabola
The vertex is a crucial point on a parabola. It represents the tip or the "turning point" of the curve. For equations in the form \[ (y - k)^2 = 4p(x - h) \] the vertex is simply at the point \((h, k)\).
In our specific example with the equation \( x = \frac{1}{4}y^2 \), it's in the form where both \( h \) and \( k \) are zero. Therefore, the vertex of this parabola is at the origin, which is the point \((0, 0)\).
The vertex is always a starting point for graphing a parabola, and it's essential to understand its location to predict the graph's behavior.
Focus of Parabola
The focus of a parabola is a point that defines its specific curve, along with the vertex and directrix. It plays a role in properties like reflection.
For a sideways opening parabola given by \[ (y - k)^2 = 4p(x - h) \] the focus is located at \((h + p, k)\).
In our equation \( x = \frac{1}{4}y^2 \), we have already identified that \( h = 0\), \( k = 0\), and \( p = 1 \). Thus, the focus of this specific parabola is at \((1, 0)\), located one unit to the right of the vertex.
The directrix is equally important and lies at \( x = -1 \), one unit to the left of the vertex.
Graphing Parabolas
Graphing parabolas begins with knowing their key components like vertex and focus. This helps sketch the curve accurately.
  • Start by plotting the vertex of the parabola. In this case, it is at \((0, 0)\).
  • Then, position the focus at \((1, 0)\) based on the calculated value of \( p \).
  • Draw the curve that opens towards the direction dictated by the parabola's orientation鈥攊n this equation, to the right.
  • Make sure the curve is symmetric about the axis passing through the focus and the vertex.
Using graphing tools, whether it's a calculator or software, can provide precision, especially in plotting additional symmetrical points to ensure accuracy and a better understanding of the parabola's shape.

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