Problem 22 - Precalculus: Mathematical for Calculus

Question

Use a graphing device to graph the parabola. $$8 y^{2}=x$$

Step-by-Step Solution

Answer

Graph the parabola \( 8y^2 = x \) on a device; it opens right from the origin.

Step 1: Understanding the Equation

The given parabola is represented by the equation \( 8y^2 = x \), which is in the form of \( y^2 = \frac{x}{8} \). This shows that the parabola opens to the right because the \( y^2 \) term is positive.

Step 2: Identify Key Components

In the equation \( y^2 = \frac{1}{8}x \), the vertex of the parabola is at the origin \((0, 0)\). The coefficient of \( x \) is \( \frac{1}{8} \), which will affect the width of the parabola.

Step 3: Graphing the Parabola

Using a graphing device, input the equation \( y^2 = \frac{1}{8}x \). The graph should appear as a parabola situated on the y-axis, with its vertex at the origin \((0, 0)\) and opening to the right. The parabolic arms will extend symmetrically on either side of the x-axis.

Key Concepts

vertex of parabola

In a parabola, the vertex is a key point where the curve changes direction. It acts like the 'nose' from where the parabola opens out. For the equation \(8 y^2 = x\), the vertex is located at the origin \((0, 0)\). Why? Because in the equation \(y^2 = \frac{1}{8}x \), you can see there are no additional constants added or subtracted from the \(x\) or \(y\). This indicates that the highest or lowest point is at \(0, 0\), which is typical for simple forms of parabolas.
Understanding the vertex is crucial because it helps in determining the parabola's position and direction on a graph. In this case, the parabola opens to the right from the vertex, as there's no \(x^2\) term restricting the opening to vertical direction. This is consistent with equations where \(y^2\) is a function of \(x\).

equation of parabola

The equation of a parabola in general can vary but has certain forms. With \(8y^2 = x\), we see that it resembles the form \(y^2 = 4px\), where \(p\) indicates how wide or narrow it is. The form helps us understand key attributes of the parabola, like its direction and how it opens.
In this equation, rearranging gives \(y^2 = \frac{1}{8}x\). Notice \(\frac{1}{8}\) here. It's a small fraction implying the parabola will open wider. The opening direction is determined by which variable the square term is attached to. If the equation were \(x^2 = 4py\), it would open vertically rather than horizontally. Recognizing these parts in an equation helps predict what the parabola will look like even before graphing.

graphing device usage

Graphing devices, like graphing calculators or software, are invaluable for visualizing equations like \(8y^2 = x\). To graph this, input it as \(y^2 = \frac{1}{8}x\), which is equivalent by rearrangement. Most graphing devices allow you to plot this directly, showing you the shape practically customize your view by zooming to see different sections.
These devices handle the heavy lifting, drawing complex graphs effortlessly and accurately. This facilitates better understanding, especially when equations become more complicated. They allow instant verification of manually plotted points or when checking characteristic points like vertexes or intersection with axes. Using graphing devices efficiently complements manual calculation skills.