Problem 74 - Intermediate Algebra

Question

Write each equation in standard form, if it is not already 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. $$ x=4 y^{2} $$

Step-by-Step Solution

Answer

The equation represents a parabola with vertex at (0, 0), opening to the right.

Step 1: Identify the Equation Type

The given equation is \( x = 4y^2 \). It resembles the equation of a parabola, which is typically of the form \( x = ay^2 + by + c \). In this case, the equation is already in a format that indicates a parabola.

Step 2: Reorganize into Standard Parabola Form

The standard form of a parabola oriented along the x-axis is \( (y - k)^2 = 4p(x - h) \). The equation \( x = 4y^2 \) can be rewritten as \( (y - 0)^2 = \frac{1}{4}(x - 0) \) by rearranging terms and comparing it with the standard form.

Step 3: Identify the Vertex and Direction

From the equation \( (y - 0)^2 = \frac{1}{4}(x - 0) \), we can tell that the vertex is at \((h, k) = (0, 0)\). Also, since the standard form \( (y - k)^2 = 4p(x - h) \) shows that \( 4p = \frac{1}{4} \), it implies \( p = \frac{1}{16} \), indicating that the parabola opens to the right.

Step 4: Graph the Parabola

To graph the parabola, start by plotting the vertex at \((0, 0)\). Since the equation indicates the parabola opens to the right, use points such as \((\pm p, \pm \sqrt{\text{plug corresponding } x})\), like \((1, 2)\) and \((1, -2)\), to plot additional points along the curve. Draw a smooth curve through these points.

Key Concepts

Parabola Graphing

Graphing a parabola may seem complex, but when broken down into steps, it becomes much more manageable. First, identifying whether the equation is a parabola is crucial. If your equation has variables in a squared form, such as \( x = 4y^2 \), it indicates a parabola.

Once you've identified the type of graph, the next step is sketching it on a coordinate plane.

• Begin by plotting the vertex, the point where the parabola changes direction.
• Since the parabola from our exercise opens to the right, plot additional points along the curve by choosing various values for \( y \) and solving for \( x \).

Remember to ensure symmetry in your graph. For this exercise, add points symmetrically around the vertex to create an accurate representation.

Once you have enough points, connect them with a smooth curve. This demonstrates the parabolic shape clearly.

Parabola Vertex

Understanding the vertex of a parabola is crucial because it's the point where the parabola changes direction. The vertex tells you a lot about the shape and orientation of the parabola. In the standard form for a horizontally oriented parabola \((y - k)^2 = 4p(x - h)\), the vertex is at \((h, k)\).

In our exercise, the problem is rewritten as \((y - 0)^2 = \frac{1}{4}(x - 0)\). This makes it evident that the vertex is located at \((0, 0)\). The vertex not only serves as a starting point for graphing but also helps you understand how far the parabola opens out from the vertex itself.

The position of the vertex along the x or y axis affects the width and direction the parabola opens. Here, knowing the vertex is at the origin simplifies the process of graphing.

Equation Standard Form

The standard form of a parabola provides a structured way to understand its different components, such as the direction it opens and its width. For parabolas that open to the sides, like in our exercise, the equation takes the form \((y-k)^2 = 4p(x-h)\).

Let's break down what each part means:

• \(h\) and \(k\) signify the vertex's coordinates.
• \(p\) determines the distance between the vertex and the focus, affecting the width of the parabola.
• The term \(4p\) directly influences how open or "tight" the parabola appears.

Therefore, rewriting equations in this standard form helps easily identify these features. It also aids in graphing parabolas accurately, as it dictates the vertex's position and the parabola's orientation.