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Magazine Chapter 4: Problem 37
Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. \(r(x)=\frac{18}{(x-3)^{2}}\)
Short Answer
Expert verified
Y-intercept: (0, 2); Vertical asymptote: x = 3; Horizontal asymptote: y = 0. No x-intercepts.
Step by step solution
01
Find the y-intercept
To find the y-intercept, set \(x = 0\) in the function and solve for \(r(x)\). This gives:\[r(0) = \frac{18}{(0-3)^{2}} = \frac{18}{9} = 2\]So, the y-intercept is \((0, 2)\).
02
Find the x-intercepts
The x-intercepts occur where the numerator is zero. Since the numerator of \(r(x) = \frac{18}{(x-3)^{2}}\) is 18, which is never zero, there are no x-intercepts.
03
Find the vertical asymptote
The vertical asymptote occurs where the denominator is zero. Set \((x-3)^2 = 0\) to solve for \(x\):\[(x-3)^{2} = 0 \x - 3 = 0 \x = 3\]Therefore, there is a vertical asymptote at \(x = 3\).
04
Determine the horizontal asymptote
For horizontal asymptotes of rational functions of the form \(\frac{a}{b}\), compare the degree of the numerator and the denominator. Since the numerator is constant (degree 0) and the denominator has degree 2, the horizontal asymptote is \(y = 0\).
05
Sketch the graph considering intercepts and asymptotes
The y-intercept is at \((0, 2)\). The vertical asymptote is a vertical line at \(x = 3\), and the horizontal asymptote is the x-axis, \(y = 0\). The function approaches these asymptotes. Plot these points and lines and make sure the curve approaches the lines but does not cross the asymptotes except at the y-intercept.
06
Confirm with a graphing device
Use a graphing calculator or software to plot \(r(x) = \frac{18}{(x-3)^{2}}\). Verify the y-intercept, and check the behavior of the function as \(x\) approaches the asymptotes. The plot should show the function approaching \(y = 0\) horizontally and \(x = 3\) vertically without crossing except at the specified intercept.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Intercepts
In mathematics, intercepts are crucial points where a function crosses the axes of a graph, specifically the x-axis and y-axis. For the function \( r(x) = \frac{18}{(x-3)^2} \), finding these intercepts gives us valuable insight into the behavior of the function.
- Y-intercept: The y-intercept is determined by setting \( x = 0 \) into the function. This gives \( r(0) = \frac{18}{9} = 2 \), which means the y-intercept is \((0, 2)\). It's where the graph crosses the y-axis.
- X-intercepts: An x-intercept occurs when the function value is zero, i.e., the numerator of the function is zero. However, in \( r(x) = \frac{18}{(x-3)^2} \), the numerator is 18 and never zero. Thus, there are no x-intercepts for this rational function.
Asymptotes
Asymptotes are lines that a graph approaches but never touches or crosses, providing a boundary for the behavior of the function's graph as it extends towards infinity. In the context of rational functions, asymptotes can be vertical, horizontal, or oblique.
- Vertical Asymptote: For \( r(x) = \frac{18}{(x-3)^2} \), the vertical asymptote is found when the denominator equals zero. Solving \((x-3)^2 = 0 \) gives \( x = 3 \). Therefore, there is a vertical asymptote at \( x = 3 \). The function approaches this vertical line but never crosses it.
- Horizontal Asymptote: The horizontal asymptote describes the behavior of \( r(x) \) as \( x \) approaches infinity. This is determined by comparing the degrees of the numerator and the denominator. Since the numerator's degree is 0 and the denominator's degree is 2, the horizontal asymptote is at \( y = 0 \). This indicates that the graph will approach the x-axis as \( x \) goes to infinity.
Graphing Rational Functions
Graphing a rational function involves piecing together information from intercepts and asymptotes to predict the graph's path and behavior. For \( r(x) = \frac{18}{(x-3)^2} \), here鈥檚 how you can graph it effectively:
- Start with the intercepts: Plot the y-intercept at \((0, 2)\). Since there are no x-intercepts, the graph does not touch the x-axis.
- Add the asymptotes: Draw the vertical asymptote as a dashed line at \( x = 3 \) and the horizontal asymptote as the x-axis \( y = 0 \).
- Sketch the graph: With these guides, sketch curves that come close to the asymptotes without crossing them. As \( x \) approaches 3 from either direction, the function value grows without bound, indicating a steep incline or decline aligning with the vertical asymptote.
- Verify with a graphing tool: Using graphing software can help validate your sketch by showing the complete behavior of the function, ensuring that it aligns with the expected intercepts and asymptotes.
