Problem 23 - Precalculus Mathematics for Calculus

Question

Use a graphing device to graph the parabola. $$x^{2}=16 y$$

Step-by-Step Solution

Answer

Graph the equation \(x^2 = 16y\) to display the upward-opening parabola with a vertex at the origin.

Step 1: Identify the Equation Type

The given equation is \(x^2 = 16y\). This is the equation of a parabola. In standard form, a vertical parabola equation is \(x^2 = 4py\), where \(p\) represents the focal length.

Step 2: Rewrite the Equation in Standard Form

We can compare \(x^2 = 16y\) with \(x^2 = 4py\). Here, \(4p = 16\). Therefore, \(p = 4\). This tells us that the focus is 4 units away from the vertex along the y-axis.

Step 3: Determine the Direction of Opening

Since the equation is \(x^2 = 4py\) with \(p = 4\), the parabola opens upwards. This is a characteristic of parabolas in this form when \(p\) is positive.

Step 4: Graphing the Parabola Using a Graphing Device

Open your graphing device or software and enter the equation \(x^2 = 16y\). This will show a parabola opening upwards. Ensure the vertex is at the origin \((0, 0)\) and plot additional points for accuracy by substituting values for \(x\) to find corresponding \(y\) values.

Step 5: Confirm the Plot

Check the graph to ensure the vertex is at \((0,0)\) and the parabola curve passes through points such as \((4,4)\), \((-4,4)\), aligning with the equation \(x^2 = 16y\). This confirms that the parabola is correctly graphed.

Key Concepts

Equation of a Parabola

The equation of a parabola is a mathematical representation of a U-shaped curve. Parabolas are commonly encountered in algebra and geometry, especially in the context of quadratic equations and projectile motion. The equation helps determine the direction and position of the parabola.

The standard form of a parabola's equation depends on the parabola's orientation:

• If the parabola opens vertically (standard form is \(x^2 = 4py\)), it is either upward or downward.
• If the parabola opens horizontally (standard form is \(y^2 = 4px\)), it can open to the left or right.

In both cases, \(p\) is a key parameter. It represents the distance from the vertex of the parabola to its focus, governing how wide or narrow the parabola appears.

Understanding the general shape and properties of a parabolic equation gives valuable insights into its geometry and real-world applications.

Standard Form of a Parabola

The standard form of a parabola's equation is crucial for easily identifying its properties and making quick graphical interpretations. In the context of vertical parabolas, the standard form is given by \(x^2 = 4py\).

To convert a given equation to this standard form, you might need to rearrange or adjust coefficients. For instance, given \(x^2 = 16y\), you can compare it with \(x^2 = 4py\) to find that \(4p = 16\). Solving for \(p\) yields \(p = 4\).

This implies a focus at \((0, 4)\) and a vertex at the origin \((0, 0)\), with the parabola opening upwards. The interpretive ease afforded by the standard form makes it invaluable for graphing and analyzing parabolic equations.

Whenever dealing with a parabola, shifting it into this standard form is a critical step for clear understanding and visualization.

Focus of a Parabola

The focus of a parabola plays a pivotal role in shaping its structure. It is a specific point located inside the parabola where all the reflected lines from the curve converge. This point is integral to understanding the properties of a parabola, especially in optical systems, as it dictates the path of reflected light or signals.

To locate the focus, refer to the standard form equation \(x^2 = 4py\) for a vertically opening parabola. Here, the distance \(p\) from the vertex (also at the origin, in this example) to the focus directly tells you the position of the focus along the y-axis.

Consider the parabola \(x^2 = 16y\);

• The equation reveals that \(4p = 16\), leading to \(p = 4\).
• Consequently, the focus is at \((0, 4)\), four units above the vertex \(\), assuming the vertex is at the origin.

Appreciating the focus's role enhances comprehension of the parabola's geometry and modeling potential applications like satellite dishes and headlight design.