Problem 49 - Algebra 2

Question

Graph each equation. $$ x^{2}=12 y $$

Step-by-Step Solution

Answer

The parabola for the given equation opens upwards with its vertex at the origin (0, 0). Some of the points on the parabola are (6,3), (-6,3), (12,12), and (-12,12).

Step 1: Identify the vertex

The vertex of the parabola for the equation in the form \(x^{2} = 4ay\) is at the origin (0, 0). Given the equation \(x^{2} = 12y\), the vertex is also (0, 0).

Step 2: Find Points that Satisfy the Equation

Select some x-values, substitute them into the equation and find the corresponding y-values. For example, if \(x = \pm 6 \), \(y = 3\). Then we get points (6,3) and (-6,3). If \(x = \pm 12\), \(y = 12\). Then we get points (12,12) and (-12,12).

Step 3: Sketch the Parabola

Draw a set of axes on a graph, plot the vertex, the points found in Step 2, and draw the curve of the parabola. The parabola will open upwards as a = 3 (which is positive).

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a significant point that represents the highest or lowest point on the curve. In the standard form equation of a parabola, \( x^2 = 4ay \), the vertex is located at the point \((h, k)\). It serves not only as a center of symmetry but also helps determine the direction in which the parabola opens.
For the given equation, \( x^2 = 12y \), it conforms to the form \( x^2 = 4ay \) with \( a = 3 \). Here, the vertex is located at the origin \((0, 0)\).
The equation itself gives us this crucial detail immediately and acts as a reference point for graphing the parabola. Thus, when analyzing a parabola, understanding and pinpointing the vertex is your first step.

• The vertex is the turning point of the parabola.
• For \( x^2 = 12y \), the vertex is at \((0, 0)\).
• The vertex helps determine the symmetry and orientation of the curve.

Graphing Quadratic Equations

Graphing quadratic equations is a process that translates the mathematical equation into a visual representation, creating a parabolic curve on the graph. To graph a quadratic equation effectively, it's crucial to identify key components and characteristics of the parabola involved, such as the vertex and direction of opening.
The equation \(x^2 = 12y\) is a vertical parabola that opens upwards because the coefficient related to the \(y\)-variable, \(a\), is positive. After determining the vertex, you find additional points by selecting certain \(x\)-values, substituting them into the equation, and solving for \(y\). This helps in creating a more accurate sketch.

• Begin with the vertex to locate the parabola's central point.
• Identify the direction it opens based on the sign of \(a\).
• Plot additional points from which the parabola will extend.

Graphing gives a visual understanding of how the curve behaves, aiding in problem-solving and real-world applications. Once these points are plotted, connect them smoothly to form the parabolic shape.

Standard Form of a Parabola

The standard form of a parabola provides a straightforward way to express its equation, generally written as \(x^2 = 4ay\) for vertical parabolas or \(y^2 = 4ax\) for horizontal parabolas.
In this situation, the given equation \( x^2 = 12y \) is a vertical parabola with the vertex at the origin. Recognizing the standard form allows us to effortlessly determine features like the vertex and the direction of opening.

• \(4a\) is the coefficient that helps in finding the directrix and focus of the parabola.
• The coefficient of \(y\) indicates whether the parabola opens upwards or downwards.

Using the standard form as a guide simplifies the process of graphing and instruction for solving various practical and theoretical problems. This form not only helps in graphing but also aids in understanding the algebraic properties of parabolas. All in all, familiarity with the standard form makes interpreting and manipulating parabolas more intuitive.