Problem 8
Question
For Problems \(1-20\), graph each rational function. Check first for symmetry, and identify the asymptotes. $$ f(x)=\frac{-x}{x^{2}-2 x-8} $$
Step-by-Step Solution
Verified Answer
The function has vertical asymptotes at \(x = 4\) and \(x = -2\), a horizontal asymptote at \(y = 0\), and is odd with origin symmetry.
1Step 1: Identify the Symmetry
To check for symmetry, determine if the function is even, odd, or neither. Substitute \(-x\) for \(x\) in the function and simplify: \[ f(-x) = \frac{-(-x)}{(-x)^2 - 2(-x) - 8} = \frac{x}{x^2 + 2x - 8} \]Since \(f(-x) = -f(x)\), the function \(f(x)\) is odd, indicating symmetry about the origin.
2Step 2: Find the Vertical Asymptotes
Vertical asymptotes occur where the denominator is zero. Set the denominator equal to zero and solve for \(x\):\[ x^2 - 2x - 8 = 0 \]To solve, factor the quadratic equation:\[ (x-4)(x+2) = 0 \]Therefore, the vertical asymptotes are at \(x = 4\) and \(x = -2\).
3Step 3: Find the Horizontal Asymptote
For rational functions, horizontal asymptotes are determined by the degrees of the polynomials in the numerator and denominator. The degree of the numerator is 1, and the degree of the denominator is 2. If the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is at \(y = 0\).
4Step 4: Graph the Function
Plot the vertical asymptotes \(x = 4\) and \(x = -2\). Draw the horizontal asymptote at \(y = 0\). Considering the symmetry about the origin, plot some test points, ensuring the function behaves as expected near the asymptotes.
Key Concepts
Symmetry in FunctionsVertical AsymptotesHorizontal AsymptotesGraphing Techniques
Symmetry in Functions
Symmetry in functions is a fascinating concept that often makes graphing easier. For our rational function, understanding symmetry helps us predict its behavior without plotting every single point. There are primarily two types of symmetry to check: even symmetry, which is symmetry around the y-axis; and odd symmetry, which is symmetry around the origin.
To check these, we substitute \(x\) with \(-x)\) in the function. For this exercise, after substitution and simplification, we get \(f(-x) = x/(x^2 + 2x - 8)\), which equals \(-f(x)\). This means the function is odd, indicating it has symmetry around the origin. This type of symmetry reflects \(f(x)\) onto \(-f(x)\), making half of our graph predictable based on the other half.
To check these, we substitute \(x\) with \(-x)\) in the function. For this exercise, after substitution and simplification, we get \(f(-x) = x/(x^2 + 2x - 8)\), which equals \(-f(x)\). This means the function is odd, indicating it has symmetry around the origin. This type of symmetry reflects \(f(x)\) onto \(-f(x)\), making half of our graph predictable based on the other half.
Vertical Asymptotes
Vertical asymptotes of a rational function indicate the x-values where the function approaches infinity, meaning the graph will not touch or cross these lines. They are found by determining where the function's denominator equals zero.
For our function \(f(x) = -x/(x^2 - 2x - 8)\), setting the denominator to zero gives us \(x^2 - 2x - 8 = 0\). Factoring this quadratic equation, we get \( (x-4)(x+2) = 0\), leading to vertical asymptotes at \(x = 4\) and \(x = -2\).
The graph will show sharp increases or decreases as it nears these x-values, resembling cliffs in the graph. Understanding and plotting these asymptotes first helps guide the overall shape and behavior of the rational function's graph.
For our function \(f(x) = -x/(x^2 - 2x - 8)\), setting the denominator to zero gives us \(x^2 - 2x - 8 = 0\). Factoring this quadratic equation, we get \( (x-4)(x+2) = 0\), leading to vertical asymptotes at \(x = 4\) and \(x = -2\).
The graph will show sharp increases or decreases as it nears these x-values, resembling cliffs in the graph. Understanding and plotting these asymptotes first helps guide the overall shape and behavior of the rational function's graph.
Horizontal Asymptotes
Horizontal asymptotes in rational functions tell us the value that the function approaches as \(x\) heads towards positive or negative infinity. For our exercise, they are identified by comparing the degrees of the polynomials in the numerator and denominator.
Here, the numerator's degree is 1 because it's simply \(-x)\), and the denominator has a degree of 2, \(x^2 - 2x - 8\). When the degree of the numerator is less than that of the denominator, like in this case, our horizontal asymptote is \(y = 0\).
This tells us that as we move far away from the origin on the graph, in both directions, the graph will get closer and closer to the x-axis, giving us a valuable guide for understanding how the edges of our graph behave.
Here, the numerator's degree is 1 because it's simply \(-x)\), and the denominator has a degree of 2, \(x^2 - 2x - 8\). When the degree of the numerator is less than that of the denominator, like in this case, our horizontal asymptote is \(y = 0\).
This tells us that as we move far away from the origin on the graph, in both directions, the graph will get closer and closer to the x-axis, giving us a valuable guide for understanding how the edges of our graph behave.
Graphing Techniques
Graphing rational functions requires combining our knowledge of symmetry and asymptotes, along with some plotting techniques. Start by drawing the axes and marking the vertical asymptotes at \(x = 4\) and \(x = -2\), ensuring the graph does not cross these lines.
Next, draw the horizontal asymptote at \(y = 0\). Analyzing symmetry, since the function is odd, you can decide the shape on one side and mirror it around the origin for the other half. Plot important points near the asymptotes to get an idea of the graph's bends and curves.
These steps help visualize how the function behaves around borders and fills in the range between these asymptotes. With these techniques, you can confidently sketch even complex rational functions.
Next, draw the horizontal asymptote at \(y = 0\). Analyzing symmetry, since the function is odd, you can decide the shape on one side and mirror it around the origin for the other half. Plot important points near the asymptotes to get an idea of the graph's bends and curves.
These steps help visualize how the function behaves around borders and fills in the range between these asymptotes. With these techniques, you can confidently sketch even complex rational functions.
Other exercises in this chapter
Problem 7
For Problems 1-10, find \(f(c)\) by (a) evaluating \(f(c)\) directly, and (b) using synthetic division and the remainder theorem. $$ f(n)=6 n^{3}-35 n^{2}+8 n-1
View solution Problem 7
Use synthetic division to determine the quotient and remainder for each problem. $$ \left(4 x^{2}+19 x-32\right) \div(x+6) $$
View solution Problem 8
Graph each of the following rational functions: $$ f(x)=\frac{x}{x-3} $$
View solution Problem 8
For Problems \(1-22\), graph each of the polynomial functions. $$ f(x)=(x-1)^{5}+2 $$
View solution