Problem 13 - Calculus Early Transcendentals: Pearson New International Edition

Question

, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts.. $$ x^{2}-y^{2}=4 $$

Step-by-Step Solution

Answer

The graph is a hyperbola with x-intercepts at (2, 0) and (-2, 0), opening left-right.

Step 1: Rearrange the Equation

The given equation is \( x^2 - y^2 = 4 \). Rearrange it if needed but, in this case, it is already in a proper form where we can analyze it directly. It resembles the equation of a hyperbola, just solve for \( y^2 \) to make identification easier: \( x^2 - 4 = y^2 \).

Step 2: Check for Symmetries

To check for symmetry, replace \( y \) with \( -y \) and \( x \) with \( -x \) in the equation. 1. Symmetry about the x-axis: Change \( y \) to \( -y \): \[ x^2 - (-y)^2 = x^2 - y^2 = 4 \]. It holds true, so the graph is symmetric about the x-axis.2. Symmetry about the y-axis: Change \( x \) to \( -x \): \[ (-x)^2 - y^2 = x^2 - y^2 = 4 \]. It holds true, so the graph is symmetric about the y-axis.3. Symmetry about the origin: Change both \( x \) to \( -x \) and \( y \) to \( -y \): \[ (-x)^2 - (-y)^2 = x^2 - y^2 = 4 \]. It holds true, so the graph is symmetric about the origin.

Step 3: Find the x-intercepts

To find the x-intercepts, set \( y = 0 \) and solve for \( x \): \[ x^2 - 0^2 = 4 \Rightarrow x^2 = 4 \Rightarrow x = \pm2 \]. The x-intercepts are \((2, 0)\) and \((-2, 0)\).

Step 4: Find the y-intercepts

To find the y-intercepts, set \( x = 0 \) and solve for \( y \): \[ 0^2 - y^2 = 4 \Rightarrow -y^2 = 4 \]. Since \( y^2 = -4 \) does not yield real solutions, there are no y-intercepts for this hyperbola.

Step 5: Plot the Hyperbola

Using the symmetry about both axes and no y-intercept, plot the hyperbola centered at the origin \((0,0)\), passing through the x-intercepts \( (2,0) \) and \( (-2,0) \). The branches open left-right along the x-axis since it's in the form \( x^2 - y^2 = c^2 \) for \( c = 2 \).

Key Concepts

Hyperbola Equation

Understanding the basic equation of a hyperbola is crucial when graphing one. The general form of a hyperbola centered at the origin is \( x^2 - y^2 = c^2 \) or \( y^2 - x^2 = c^2 \). In our case, the equation given is \( x^2 - y^2 = 4 \), which matches the first form.
This means the hyperbola opens to the right and left along the x-axis. The constant on the right side of the equation, 4, determines the distance between the center of the hyperbola and the vertices on the x-axis. By setting \( c = \sqrt{4} = 2 \), we can easily identify that the equation represents a hyperbola with vertices located at \((2, 0)\) and \((-2, 0)\).
Recognizing this form allows for a more straightforward approach to graphing this shape and predicting its path.

Symmetry in Graphs

A hyperbola often exhibits symmetry, and identifying this can simplify graphing tasks. Symmetry checks involve substituting \( x \) and \( y \) with their negative counterparts:

• Symmetry about the x-axis: Replacing \( y \) with \( -y \) yields \( x^2 - (-y)^2 = x^2 - y^2 = 4 \), showing symmetry about the x-axis.
• Symmetry about the y-axis: By replacing \( x \) with \( -x \), we get \( (-x)^2 - y^2 = x^2 - y^2 = 4 \), confirming symmetry about the y-axis.
• Symmetry about the origin: Changing both \( x \) and \( y \) to their negatives gives \((-x)^2 - (-y)^2 = x^2 - y^2 = 4\), proving symmetry about the origin.

Discovering these symmetries helps in quickly sketching the hyperbola as well as understanding its basic structure.

X-intercepts and Y-intercepts

Intercepts provide points through which the hyperbola crosses the axes, essential for accurate graphing. To find the x-intercepts, we set \( y = 0 \) in the equation:

• \( x^2 - 0^2 = 4 \rightarrow x^2 = 4 \)
• This simplifies to \( x = \pm 2 \)

Thus, the x-intercepts are at \( (2,0) \) and \( (-2,0) \).
For y-intercepts, set \( x = 0 \):

• \( 0^2 - y^2 = 4 \rightarrow -y^2 = 4 \)
• This equation leads to no real solutions for \( y \)

Hence, the hyperbola does not have any y-intercepts. This information helps in plotting and shaping the overall graph.

Steps in Plotting Graphs

Graphing a hyperbola involves a few straightforward steps that start with understanding its shape from the equation's form. Here’s a simple process to follow:

• Identify the form: Recognize the equation’s structure to predict the hyperbola's opening direction and spread, such as \( x^2 - y^2 = 4 \) which opens left and right along the x-axis.
• Determine symmetry: Use symmetry about the axes to mirror points, simplifying plot creation.
• Find intercepts: Calculate where the hyperbola crosses axes by determining x-intercepts and identifying any lack of y-intercepts.
• Sketch the curve: Use symmetry and intercepts to draft the curve, ensuring the branches open correctly according to the hyperbola's form and features.

These logical steps guarantee a clear and accurate graph, offering a visual representation of the algebraic function.