Problem 29 - The Calculus with Analytic Geometry

Question

Draw a sketch of the graph of each of the following equations: (a) \(y=\sqrt{2 x}\) (b) \(y=-\sqrt{2 x}\) (c) \(y^{2}=2 x\)

Step-by-Step Solution

Answer

The graph for \(y = \sqrt{2x}\) is a rightward-opening parabola for positive\(y\), \(y = -\sqrt{2x}\) for negative \(y\), and \(y^2 = 2x\) combines them.

Step 1: Understanding the equations

First, understand the form of each given equation to know what type of graph to expect. For each equation, identify that they are parabolas.

Step 1: Sketch the graph of (a) \(y = \sqrt{2x}\)

This represents the positive square root of a scaled version of \(x\). It is a parabola opening to the right. To sketch it, select a few values of \(x\) (e.g., 0, 1, 2, 4) and compute the corresponding \(y\) values: \(x = 0\) -> \(y = 0\) \(x = 1\) -> \(y = \sqrt{2} \approx 1.414\) \(x = 2\) -> \(y = 2\) \(x = 4\) -> \(y = 2\sqrt{2} \approx 2.828\). Plot these points and draw a smooth curve through them that opens to the right.

Step 2: Sketch the graph of (b) \(y = -\sqrt{2x}\)

This is similar to part (a) but represents the negative square root. It is a parabola opening downward/rightward. Use the same \(x\) values (e.g., 0, 1, 2, 4) and compute: \(x = 0\) -> \(y = 0\) \(x = 1\) -> \(y = -\sqrt{2} \approx -1.414\) \(x = 2\) -> \(y = -2\) \(x = 4\) -> \(y = -2\sqrt{2} \approx -2.828\). Plot these points and draw a smooth curve through them that opens downward/rightward.

Step 3: Sketch the graph of (c) \(y^2 = 2x\)

This represents a parabola opening to the right which is equivalently seen as the union of the graphs in parts (a) and (b). Combine the points computed in Steps 1 and 2. Thus, both the upper and lower parts form: \(x = 0\) -> \(y = 0\) \(x = 1\) -> \(y = \pm \sqrt{2} \approx \pm 1.414\) \(x = 2\) -> \(y = \pm 2\) \(x = 4\) -> \(y = \pm 2\sqrt{2} \approx \pm 2.828\). Now, plot and smoothly connect these points to form the complete parabola.

Key Concepts

Parabolic Equations

Understanding parabolic equations is crucial for graphing parabolas. A standard parabolic equation can be represented in the form of \(y = ax^2 + bx + c\). However, the given set of exercises involve equations of different forms: \(y = \sqrt{2x}\), \(y = -\sqrt{2x}\), and \(y^2 = 2x\). These forms indicate that the parabolas will open either horizontally (left/right) or vertically (upward/downward). Recognizing these forms helps you predict the shape of the graph even before plotting points. For instance, \(y = \sqrt{2x}\) shows a positive square root, meaning the graph will open to the right. On the other hand, \(y = -\sqrt{2x}\) illustrates a negative square root, leading to a downward/rightward opening. Finally, the equation \(y^2 = 2x\) is a standard sideways parabola opening to the right. It actually combines the positive and negative square roots, covering both the upper and lower branches of the parabola. Recognizing these variations is essential in understanding how different equations modify the direction and shape of parabolas.

Positive and Negative Square Roots

When dealing with square-root functions in parabolic equations, it’s crucial to understand the impact of positive and negative square roots on the graph. A square-root function generally involves an expression under the radical sign, like \(\sqrt{2x}\).

Here are some key points:

• \(y = \sqrt{2x}\) involves taking the positive square root only. This graph opens to the right because \(y\) increases as \(x\) increases.
• \(y = -\sqrt{2x}\) involves taking the negative square root, meaning the graph reflects across the x-axis compared to \(y = \sqrt{2x}\).

For these cases, choose a few values of \(x\) (like 0, 1, 2, and 4) and compute the corresponding \(y\) values. For example:

• When \(x = 1\), \(y = \sqrt{2} \approx 1.414\) and \(y = -\sqrt{2} \approx -1.414\).
• When \(x = 4\), \(y = 2\sqrt{2} \approx 2.828\) and \(y = -2\sqrt{2} \approx -2.828\).

This helps in directly plotting and sketching the corresponding parabolic graphs accurately.

Therefore, square roots define the parabolic curves by deciding whether they will be above or below the x-axis in their respective portions.

Graph Sketching

Graphing parabolas requires plotting points and understanding their relations. To sketch a parabola:

• Identify the vertex, the turning point of the graph. For our examples, the vertex is at the origin \(x = 0, y = 0\).
• Choose key points on one or both sides of the vertex. Use simple values for \(x\) to calculate corresponding \(y\) values.
• Plot those points on a Cartesian plane.

When graphing \(y = \sqrt{2x}\), you might plot:

• Point (0, 0)
• Point (1, \sqrt{2} \approx 1.414)
• Point (4, 2\fix2) \approx 2.828)

Connect these points with a smooth curve opening to the right.

For \(y = -\sqrt{2x}\):

• Point (0, 0)
• Point (1, -\sqrt{2} \approx -1.414)
• Point (4, -2\sqrt{2} \approx -2.828)

Connect these points with a curve opening downward/rightward.
Finally, combine both curves for \(y^2 = 2x\), plotting both positive and negative roots at each \(x\) value. Thus, you create a complete parabola that extends both above and below the x-axis. Sketching parabolas involves recognizing patterns, understanding transformations, and accurately plotting points to form smooth curves.