Problem 105 - A Graphical Approach to College Algebra

Question

In Exercises \(97-108,\) graph by hand the equation of the circle or the parabola with a horizontal axis. $$x=2 y^{2}$$

Step-by-Step Solution

Answer

The parabola opens to the right with vertex at (0,0).

Step 1: Recognize the Equation Type

The given equation is \( x = 2y^2 \), which follows the form of a parabola. In this case, the parabola opens horizontally.

Step 2: Identify the Orientation and Opening

For a parabola in the form \( x = ay^2 \), the parabola opens horizontally. Since \( a = 2 \) is positive, the parabola opens to the right.

Step 3: Determine the Vertex of the Parabola

The vertex of the equation \( x = ay^2 \) is at the origin \((0, 0)\) since there is no \( y \) or \( x \) displacement term.

Step 4: Plot Key Points

Select several values for \( y \) to find corresponding \( x \) values. For instance, if \( y = 0 \), then \( x = 0 \). If \( y = 1 \), then \( x = 2 \), and if \( y = -1 \), then \( x = 2 \). Similarly, choose other \( y \) values like \( 2 \) and \( -2 \) to get further points.

Step 5: Draw the Parabola

Plot the vertex and the points calculated in Step 4 on the coordinate plane. Then, sketch a smooth curve through these points to represent the parabola.

Key Concepts

Equation Recognition

Recognizing the type of equation you're working with is crucial for properly graphing it. For our given equation, \( x = 2y^2 \), it recognizes the form of a parabola since it contains a squared term \( y^2 \). The general form of a vertically oriented parabola is \( y = ax^2 \), whereas \( x = ay^2 \) indicates a horizontal orientation. This recognition helps in predicting how the graph would look on a coordinate plane.
Identifying the equation type is the foundation of graphing, as it tells you the basic shape and direction of your graph. In exams or homework exercises, training to quickly recognize equations can help you efficiently complete graphing tasks with confidence.

Graphing Techniques

Graphing a parabola involves a series of careful steps. Firstly, always identify the vertex, the starting point of your graph. For \( x = ay^2 \), the vertex is at \((0, 0)\), the origin. Next, use a selection of values for \( y \). Substituting these into the equation, calculate the corresponding \( x \) values. This gives you crucial points to plot:

• When \( y = 0 \), \( x = 0 \).
• When \( y = 1 \) and \( y = -1 \), \( x = 2 \).
• When \( y = 2 \) and \( y = -2 \), \( x = 8 \).

Once you have plotted these points on the coordinate plane, you can sketch a smooth curve through them to visualize the parabola. This hands-on technique allows you to manually confirm the graph's shape and accuracy.

Parabola Orientation

Orientation is an important detail when graphing parabolas. It's the direction in which a parabola opens. The equation \( x = 2y^2 \) signifies a horizontally oriented parabola. For parabolas, the coefficient of the squared term determines the direction:

• If \( a > 0 \), the parabola opens to the right for equations of the form \( x = ay^2 \).
• If \( a < 0 \), it opens to the left.

Thus, since \( a = 2 \) in our problem, the parabola opens to the right. Such orientation specifies how a parabola spreads on the graph, affecting its visualization on the coordinate plane.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we graph equations. Familiarity with it is essential for plotting any function, including parabolas. It consists of two axes:

• The horizontal \( x \)-axis
• The vertical \( y \)-axis

These axes divide the plane into four quadrants. Having your points carefully plotted onto this plane allows you to see the parabola's path. In this particular case, since the parabola is horizontally oriented, expect significant horizontal stretching.
Using the coordinate plane effectively means understanding axis labels, scale, and ensuring your plotted points accurately represent your calculations. Mastery of this concept is key to successful graphing in mathematics.