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For the function \( F(x)=\frac{x^{2}-6}{x^{2}} \), (a) graph the rational function

Ask by Boone Logan. in the United States
Mar 16,2025

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To graph the function \( F(x) = \frac{x^2 - 6}{x^2} \): 1. **Rewrite the Function:** \[ F(x) = 1 - \frac{6}{x^2} \] 2. **Domain:** - All real numbers except \( x = 0 \). 3. **Intercepts:** - \( x \)-intercepts at \( x = \sqrt{6} \) and \( x = -\sqrt{6} \). - No \( y \)-intercept. 4. **Asymptotes:** - Vertical asymptote at \( x = 0 \). - Horizontal asymptote at \( y = 1 \). 5. **Symmetry:** - The function is even, symmetric about the \( y \)-axis. 6. **Key Points:** - \( F(1) = -5 \) - \( F(2) = -0.5 \) - \( F(3) \approx 0.333 \) 7. **Graph Characteristics:** - Approaches the vertical asymptote at \( x = 0 \) from both sides. - Approaches the horizontal asymptote \( y = 1 \) as \( |x| \) increases. - Crosses the \( x \)-axis at \( \sqrt{6} \) and \( -\sqrt{6} \). 8. **Sketch the Graph:** - Draw vertical and horizontal asymptotes. - Plot intercepts and key points. - Mirror points across the \( y \)-axis due to symmetry. - Ensure the graph levels off near \( y = 1 \) for large \( |x| \) and approaches \( y = -\infty \) near \( x = 0 \). This summary provides all necessary information to accurately sketch the graph of the given function.

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To graph the rational function \( F(x) = \frac{x^2 - 6}{x^2} \), start by identifying key features. The function can be simplified to \( F(x) = 1 - \frac{6}{x^2} \). This indicates that as \( x \) approaches 0, \( F(x) \) heads towards negative infinity, indicating a vertical asymptote at \( x = 0 \). Additionally, the horizontal asymptote is \( y = 1 \), as \( x \) approaches ±∞. So, plot points around these features to create a smooth curve that approaches the asymptotes. Next, look for intercepts: for the x-intercept, set \( F(x) = 0 \) which gives \( x^2 - 6 = 0 \) leading to \( x = ±\sqrt{6} \), and for the y-intercept, evaluate \( F(0) \), which confirms the vertical asymptote. Plotting these points and asymptotes provides a clear picture of the function’s behavior across the coordinate plane!

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