Trigonometry (11th Edition)
11th Edition
ISBN: 9780134217437
Author: Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher: PEARSON
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I need help how to solve this please. I am so confused. Round to three decimal places. Thank you.
![### Using a Graphing Utility to Analyze a Function
#### Function Analysis
The given function is:
\[ y = \frac{2}{1-x^2} \]
#### Step 1: Construct a Table of Values
Using a graphing utility, create a table of values for the function. Make sure to round your answers to three decimal places. Here's an example of how to structure your table:
\[
\begin{array}{c|c}
x & y = \frac{2}{1 - x^2} \\
\hline
-3 & \\
-2 & \\
-1 & \\
0 & \\
1 & \\
\end{array}
\]
#### Step 2: Sketch the Graph of the Function
Below are four options for the potential graph of the function \( y = \frac{2}{1-x^2} \). Select the correct graph by analyzing the function's characteristics.
**Graph Options:**
1. **First graph:**
- y-axis range from -4 to 4.
- x-axis range from -4 to 4.
- Graph shows a peak around \( x = 0 \) and falls towards \( y = 1 \) as \( x \) moves away from 0.
2. **Second graph:**
- Similar axes range as the first graph.
- Graph shows a sharp peak at \( x = 1 \) and \( x = -1 \) with steep drops.
3. **Third graph:**
- Similar axes range.
- Graph dips with a valley at \( y = -2 \).
4. **Fourth graph:**
- Similar axes range.
- Shows a valley dipping below \( y = 0 \) but more prominently.
#### Step 3: Identify Asymptotes
Determine the location of any asymptotes for the given function.
- **Vertical Asymptote:**
Vertical asymptotes occur where the function is undefined. For the function \( y = \frac{2}{1-x^2} \), the denominator \( 1 - x^2 \) equals zero at \( x = \pm 1 \), leading to vertical asymptotes at these points.
\[ \text{Vertical asymptote:} \ x = \pm 1 \]
- **Horizontal Asymptote:**
As \( x \) approaches infinity](https://content.bartleby.com/qna-images/question/8bee126f-2352-405d-8ef2-fdb48e669f56/f0eca2e7-4bbf-40d9-980a-713cdcbe7c90/91ve1j_processed.jpeg)
Transcribed Image Text:### Using a Graphing Utility to Analyze a Function
#### Function Analysis
The given function is:
\[ y = \frac{2}{1-x^2} \]
#### Step 1: Construct a Table of Values
Using a graphing utility, create a table of values for the function. Make sure to round your answers to three decimal places. Here's an example of how to structure your table:
\[
\begin{array}{c|c}
x & y = \frac{2}{1 - x^2} \\
\hline
-3 & \\
-2 & \\
-1 & \\
0 & \\
1 & \\
\end{array}
\]
#### Step 2: Sketch the Graph of the Function
Below are four options for the potential graph of the function \( y = \frac{2}{1-x^2} \). Select the correct graph by analyzing the function's characteristics.
**Graph Options:**
1. **First graph:**
- y-axis range from -4 to 4.
- x-axis range from -4 to 4.
- Graph shows a peak around \( x = 0 \) and falls towards \( y = 1 \) as \( x \) moves away from 0.
2. **Second graph:**
- Similar axes range as the first graph.
- Graph shows a sharp peak at \( x = 1 \) and \( x = -1 \) with steep drops.
3. **Third graph:**
- Similar axes range.
- Graph dips with a valley at \( y = -2 \).
4. **Fourth graph:**
- Similar axes range.
- Shows a valley dipping below \( y = 0 \) but more prominently.
#### Step 3: Identify Asymptotes
Determine the location of any asymptotes for the given function.
- **Vertical Asymptote:**
Vertical asymptotes occur where the function is undefined. For the function \( y = \frac{2}{1-x^2} \), the denominator \( 1 - x^2 \) equals zero at \( x = \pm 1 \), leading to vertical asymptotes at these points.
\[ \text{Vertical asymptote:} \ x = \pm 1 \]
- **Horizontal Asymptote:**
As \( x \) approaches infinity
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