Problem 31
Question
Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$f(x)=\frac{1}{x-2}$$
Step-by-Step Solution
Verified Answer
The graph of the function \(f(x)=\frac{1}{x-2}\) has a vertical asymptote at \(x = 2\) and a horizontal asymptote at \(y = 0\). It crosses the y-axis at \(y = -0.5\) and does not pass through the x-axis.
1Step 1: Identify Vertical Asymptotes
Vertical asymptotes occur at the x-values where the denominator of the rational function is equal to 0. For this function \(f(x)=\frac{1}{x-2}\), the denominator \(x - 2\) equals zero at \(x = 2\). Therefore, \(x = 2\) is the vertical asymptote.
2Step 2: Identify Horizontal Asymptotes
Horizontal asymptotes are determined by examining the degrees of the numerator and denominator of the rational function. In \(f(x)=\frac{1}{x-2}\), the degree of the denominator \(x - 2\) is 1 and the degree of the numerator 1 is 0. When the degree of the denominator is greater than the numerator, the x-axis (thus \(y = 0\)) is the horizontal asymptote.
3Step 3: Identify Intercepts
The x-intercept is the value of x where the function crosses the x-axis, i.e., where \(y = 0\). For the function \(f(x)=\frac{1}{x-2}\), it never equals to zero, thus there are no x-intercepts. The y-intercept is the value of the function at \(x = 0\), hence, \(f(0)=\frac{1}{0-2} = -0.5\), so the y-intercept is at \(y = -0.5\).
4Step 4: Sketch the Curve
The function \(f(x)=\frac{1}{x-2}\) has a vertical asymptote at \(x = 2\) and a horizontal asymptote at \(y = 0\). It also does not pass through the x-axis but does cross the y-axis at \(y = -0.5\). With these critical points, the graph can be sketched accordingly. On the left side of the vertical asymptote, the function tends towards positive infinity as \(x\) decreases, while it tends towards negative infinity as \(x\) increases on the right side of the vertical asymptote.
Key Concepts
Vertical AsymptotesHorizontal AsymptotesIntercepts
Vertical Asymptotes
In a rational function, vertical asymptotes are the lines that the graph tends towards but never actually touches. These occur when the denominator of the rational function is zero. This means there is a break, or "hole," in the graph at that point. It's important to remember that the function will not have any values at this x-value.
For the function \(f(x)=\frac{1}{x-2}\), we find the vertical asymptote by setting the denominator equal to zero: \(x-2=0\). Solving for \(x\) gives \(x=2\). Here, the graph will approach this line infinitely close but will never reach it. This creates the vertical asymptote at \(x=2\).
Graphically, as \(x\) approaches 2 from the left, the value of \(f(x)\) skyrockets to infinity. Conversely, approaching from the right, \(f(x)\) plunges to negative infinity. This indicates a significant change in the graph's behavior at this asymptote.
For the function \(f(x)=\frac{1}{x-2}\), we find the vertical asymptote by setting the denominator equal to zero: \(x-2=0\). Solving for \(x\) gives \(x=2\). Here, the graph will approach this line infinitely close but will never reach it. This creates the vertical asymptote at \(x=2\).
Graphically, as \(x\) approaches 2 from the left, the value of \(f(x)\) skyrockets to infinity. Conversely, approaching from the right, \(f(x)\) plunges to negative infinity. This indicates a significant change in the graph's behavior at this asymptote.
Horizontal Asymptotes
Horizontal asymptotes occur when the function gradually levels off as \(x\) moves towards infinity in either direction. They give us a snapshot of the function's long-term behavior.
To find horizontal asymptotes, compare the degrees of the numerator and the denominator. For \(f(x)=\frac{1}{x-2}\), the denominator has a degree of 1, and the numerator has a degree of 0.
If the degree of the numerator is less than that of the denominator, the horizontal asymptote is found along the x-axis, or \(y=0\). This means as \(x\) gets very large or very small, the function transforms into increasingly tiny values, smoothly approaching y=0.
To find horizontal asymptotes, compare the degrees of the numerator and the denominator. For \(f(x)=\frac{1}{x-2}\), the denominator has a degree of 1, and the numerator has a degree of 0.
If the degree of the numerator is less than that of the denominator, the horizontal asymptote is found along the x-axis, or \(y=0\). This means as \(x\) gets very large or very small, the function transforms into increasingly tiny values, smoothly approaching y=0.
Intercepts
Intercepts are points where the function crosses the axes, giving us crucial information about the graph's location in the coordinate plane. Intercepts can be of two types - x-intercepts and y-intercepts.
X-intercepts are found by setting the function equal to zero and solving for \(x\). In the case of \(f(x)=\frac{1}{x-2}\), there's no value of \(x\) that will make the function zero since there is no term in the numerator other than 1. Thus, there are no x-intercepts for this function.
Y-intercepts are determined by evaluating the function at \(x=0\). Substitute \(0\) into the function: \(f(0)=\frac{1}{0-2}=-0.5\). Therefore, the graph crosses the y-axis at \(y=-0.5\), marking it as the function's sole intercept. This intercept helps us recognize and locate the function's position within the graphical plane.
X-intercepts are found by setting the function equal to zero and solving for \(x\). In the case of \(f(x)=\frac{1}{x-2}\), there's no value of \(x\) that will make the function zero since there is no term in the numerator other than 1. Thus, there are no x-intercepts for this function.
Y-intercepts are determined by evaluating the function at \(x=0\). Substitute \(0\) into the function: \(f(0)=\frac{1}{0-2}=-0.5\). Therefore, the graph crosses the y-axis at \(y=-0.5\), marking it as the function's sole intercept. This intercept helps us recognize and locate the function's position within the graphical plane.
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