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Magazine Chapter 1: Problem 19
Find an appropriate graphing software viewing window for the given function and use it to display its graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function. $$f(x)=\frac{x^{2}+2}{x^{2}+1}$$
Short Answer
Expert verified
Use a viewing window of \(-10 \leq x \leq 10\) and \(0 \leq y \leq 2.5\) to display the function's graph.
Step by step solution
01
Analyze the Function
Start by analyzing the given function \( f(x) = \frac{x^2 + 2}{x^2 + 1} \). Notice that it is a rational function and analyze any noteworthy points or behavior. For large values of \(x\), both the numerator and denominator become \(x^2\). Calculate the horizontal asymptote by dividing the leading coefficients: The horizontal asymptote is \( y = 1 \).
02
Check for Symmetry and Intercepts
Determine the symmetry of the function. Since \( f(-x) = f(x) \), the function is even, meaning it is symmetric about the y-axis. Next, find the y-intercept by evaluating \( f(0) = \frac{2}{1} = 2 \). Check if there are any x-intercepts by solving \( f(x) = 0 \) which leads to no solution as the numerator never becomes zero.
03
Identify Vertical Asymptotes
Find any potential vertical asymptotes by setting the denominator equal to zero: \( x^2 + 1 = 0 \). Since this equation has no real solutions, there are no vertical asymptotes.
04
Define an Appropriate Viewing Window
Based on the analysis, consider setting up a viewing window to capture the behavior of the function. Since the function is symmetric and no vertical asymptotes exist, select a viewing window in \( x \) from approximately \(-10\) to \(10\) and in \( y \) from \(0\) to \(2.5\), to observe the rise toward the horizontal asymptote and the maximum y-intercept point.
05
Graph the Function
Use graphing software like Desmos, Geogebra, or a graphing calculator with the selected viewing window. Plot the function and ensure the graph reflects the characteristics noted: a horizontal asymptote at \( y = 1 \), even symmetry, maximum at (0,2), no x-intercepts nor vertical asymptotes.
06
Evaluate Graph Accuracy
After plotting, verify that the graph properly illustrates the overall behavior of the function. Ensure the symmetry about the y-axis is visible and that it approaches \( y = 1 \) symmetrically for large positive and negative \( x \) values. Adjust the window if important features are not clearly visible.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Rational Functions
Rational functions are fractions that contain polynomials in both the numerator and the denominator. They offer a range of interesting and complex behaviors when graphed. In the function \(f(x) = \frac{x^2 + 2}{x^2 + 1}\), understanding the components helps. The polynomial \(x^2 + 2\) is in the numerator and \(x^2 + 1\) in the denominator.
Key points about rational functions:
Key points about rational functions:
- They can have zeros where the numerator is zero.
- They may have undefined points (not present here) if the denominator equals zero.
- These functions might show horizontal or vertical asymptotic behavior.
Exploring Asymptotes in Graphs
Asymptotes are lines that a graph approaches but never reaches. Rational functions commonly exhibit this feature. For \(f(x) = \frac{x^2 + 2}{x^2 + 1}\), a horizontal asymptote appears at \(y = 1\).
Here's how asymptotes work:
Here's how asymptotes work:
- Horizontal Asymptotes: These occur because the degrees of the numerator and denominator are equal. The ratio of leading coefficients determines it. Here, both are 1.
- Vertical Asymptotes: They arise where the denominator is zero without cancellation from the numerator. This function has no real x values that make the denominator zero, thus no vertical asymptotes.
Recognizing Symmetry in Functions
Symmetry helps simplify graphing by showing patterns in a function. For \(f(x) = \frac{x^2 + 2}{x^2 + 1}\), symmetry is easy to spot鈥攊t's an even function. Here鈥檚 how to spot symmetry:
- Even Symmetry: A function is even if \(f(-x) = f(x)\), meaning it reflects over the y-axis. This is confirmed by observing the function's terms are unchanged when \x is replaced with \-x\.
- Odd Symmetry: Occurs if \(f(-x) = -f(x)\), causing point symmetry about the origin. This is not applicable for this function.
Using Graphing Software Effectively
Graphing software like Desmos or Geogebra is invaluable for visualizing functions. It helps when setting viewing windows that show all key function behaviors.
Power of graphing tools:
Power of graphing tools:
- Easy Adjustment: Change the viewing window as needed to ensure all features are visible, such as intercepts, maximum points, and asymptotes.
- Visual Verification: Check theoretical predictions like symmetry and limits graphically for added confirmation.
- Detailed Analysis: Examine parts of the graph in-depth by zooming in on areas of interest.
Analyzing Function Behavior
Analyzing a function's behavior involves studying how it reacts to different inputs. For \(f(x) = \frac{x^2 + 2}{x^2 + 1}\), considering behavior across a range is crucial.
Here's what to do:
Here's what to do:
- Intercepts: The y-intercept is found by evaluating f(0), giving a value of 2. No x-intercepts exist as the numerator doesn鈥檛 become zero.
- Limits: Evaluate limits as x approaches infinity or negative infinity, confirming the horizontal asymptote at y = 1.
- Extreme Points: Consider where the function may reach maximum or minimum values on its domain.
