Problem 7 - Intermediate Algebra

Question

The graph of each equation is a parabola. Find the vertex of the parabola and then graph it. $$x=3 y^{2}$$

Step-by-Step Solution

Answer

The vertex of the parabola is (0, 0).

Step 1: Convert Equation to Standard Form

The given equation is \( x = 3y^2 \). To find the vertex, we need to rewrite it in the form \( (y-k)^2 = 4p(x-h) \). This form represents parabolas that open horizontally. In the current form, we have \( x = 3(y-0)^2 \), which is already in the standard form, where \( h = 0 \), \( k = 0 \), and \( 4p = \frac{1}{3} \).

Step 2: Identify the Vertex

From the equation \( x = 3(y-0)^2 \), we can directly see that the vertex of the parabola is \((h, k) = (0, 0)\).

Step 3: Graph the Parabola

To graph the parabola given by \( x = 3y^2 \), note the direction and shape: the parabola opens to the right because \( y \) is squared and \( x \) is on the left side. Plot the vertex at the origin (0,0). Choose a few values for \( y \), such as 1 and -1, and compute the corresponding \( x \) values. For example, when \( y = 1 \), \( x = 3(1)^2 = 3 \), giving the point (3, 1). Plot these points and use symmetry to complete the graph.

Key Concepts

Parabola Graphing

Graphing a parabola can be a fun and insightful process. A parabola is a curve that appears like a U or an inverted U shape. It can open upwards, downwards, left, or right.
The orientation depends on the variable that is squared. For instance, in the exercise with the equation \( x = 3y^2 \), the parabola opens horizontally to the right because the variable \( y \) is squared.To graph this parabola:

• Start by plotting the vertex. In this case, the vertex is at the origin (0, 0).
• Next, choose some values for the variable \( y \), compute corresponding \( x \) values using the equation.
• For more points, choose both positive and negative \( y \) values. For example: when \( y = 1 \), \( x = 3(1)^2 = 3 \), resulting in the point (3, 1).
• Symmetry can help in filling the entire parabola. The points will mirror across the axis of symmetry, which is the line passing through the vertex in this case.

Graphing parabolas using these steps offers a visual representation of quadratic relationships.

Standard Form of a Parabola

Understanding the standard form of a parabola is essential for graphing. Parabolas can open up, down, left, or right depending on their equations. The standard form for parabolas that open horizontally (left or right) is \( (y-k)^2 = 4p(x-h) \). This form helps in identifying the vertex and orientation of the parabola easily.
For the equation \( x = 3y^2 \), it is transformed to fit the standard form, resulting in \( (y-0)^2 = \frac{1}{3}(x-0) \). Here:

• \( h = 0 \), \( k = 0 \), indicating the vertex at (0,0)
• The coefficient \( 4p = \frac{1}{3} \), determining the width and direction of the parabola

The standard form gives you clarity on how the parabola is positioned on the graph, making it a powerful tool for understanding the graph and behavior of the equation.

Graphing Techniques

Having some reliable graphing techniques lets you sketch parabolas quickly and accurately. Besides plotting the vertex and a few points, here are some best practices to keep in mind while graphing:

• Symmetry: Remember that the parabola is symmetric about the axis of symmetry. Use this property to plot mirrored points, reducing the amount of calculation needed.
• Direction: Check the coefficient of the squared term. If it is positive, the parabola opens upwards (for \( y \)) or to the right (for \( x \)). If negative, it opens downwards or to the left.
  
• Choosing the right values: Picking integers around the vertex can simplify calculations. For larger or smaller parabolas, adjust chosen inputs accordingly to accurately represent the curve.
• Use technology: Graphing calculators or software can confirm your sketch and provide deeper insights into transformations and specific values.

Utilize these techniques to enhance your graphing process, ensuring accuracy and a deeper understanding of parabola shapes. In no time, graphed parabolas will align perfectly with their algebraic descriptions.