Problem 19 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

Graph the parabola \( x^2 = 16y \), opening upwards from the origin.

Step 1: Recognize the equation

The given equation is a parabola in the form of \( x^2 = 4py \), where \( p \) is the parameter that represents the distance from the vertex to the focus. Our equation \( x^2 = 16y \) can be rewritten as \( x^2 = 4(4)y \), noticing that here \( 4p = 16 \), so \( p = 4 \). This indicates that the parabola opens upwards.

Step 2: Identify key components

From the equation \( x^2 = 16y \), identify important features: the vertex (0,0), the focus (0, 4), and the directrix \( y = -4 \). These will help in accurately plotting the parabola.

Step 3: Use a graphing device

Using a graphing calculator or software, input the equation \( x^2 = 16y \). The graph will display the parabola opening upwards from the origin with a vertex at (0,0) and crossing points at approximately \((4,1)\) and \((-4,1)\).

Step 4: Analyze the graph

Examine the graphed parabola to confirm that it is symmetric with respect to the y-axis, opens upwards, and contains the vertex at (0,0) and passes through the expected points.

Key Concepts

Graphing Parabola

Graphing a parabola is a visual method to understand its shape and important features. To graph the given equation \( x^2 = 16y \), you first need to rewrite it in a recognizable form, specifically, the standard form \( x^2 = 4py \). This equation is particularly useful in identifying the orientation and key points of the parabola. Common graphing tools include graphing calculators or software programs designed to handle equations. These devices allow the user to input the equation directly and instantly visualize the parabola.
Here’s a step-by-step approach to graphing this type of parabolic equation:

• Rewrite the equation into a standard form if needed, as done with \( x^2 = 4(4)y \).
• Identify the direction: Here the parabola opens upwards because the coefficient of \( y \) is positive.
• Use either graphing software or a graphing calculator to input the equation and see the parabola plotted live.
• Check that the graph exhibits the expected symmetry and structure: for this equation, it should appear symmetric along the y-axis.

Understanding the graph intuitively helps in analyzing related problems and provides visual confirmation of all calculated elements.

Vertex of Parabola

The vertex of a parabola is a critical feature, often representing its peak or low point, depending on its orientation. In the case of the equation \( x^2 = 16y \), the vertex is located at the origin, (0,0). When graphing such equations, the vertex is generally the starting point for plotting the parabola as it represents the turning point of the curve.
Important details to keep in mind about the vertex:

• It is determined directly from the equation in the form \( x^2 = 4py \), where the vertex is \((0,0)\) unless translated.
• The vertex serves as a point of symmetry, meaning the parabola is mirrored horizontally across this point.
• In practice, the vertex helps in defining the opening of the parabola. If \( p \) is positive, like here, the parabola opens upward; if \( p \) were negative, it would open downward.

This foundational aspect greatly assists in structuring the rest of the graphing process and confirms the orientation and position relative to the coordinate plane.

Focus and Directrix of Parabola

The focus and directrix are essential components that contribute to the definition and structure of a parabola. In the parabola defined by \( x^2 = 16y \), these elements play crucial roles.
The focus of a parabola is a point from which distances are measured in defining the curve. For the equation \( x^2 = 16y \), we determined earlier that \( 4p=16 \), so \( p=4 \). Thus, the focus is located at (0,4). This is calculated as the point \( (0, p) \) when the parabola is oriented vertically.
The directrix is a line that is perpendicular to the axis of symmetry of the parabola. For this particular equation, taking \( y = -4 \) as the directrix gives us the location based on \( p \). Such a configuration makes the curve equidistant from the focus and directrix at any horizontal position along the parabola.

• The focus provides the focal point towards which all parabolic paths converge.
• The directrix acts as a baseline from which distances to the curve's points are considered equidistant to the focus.

Understanding these components is crucial in comprehensively analyzing and interpreting the parabola's characteristics and behavior.