College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter4: Polynomial And Rational Functions
Section4.6: Rational Functions
Problem 99E
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![### Graphing a Rational Function Using the Six-Step Procedure
To graph the rational function \( f(x) = \frac{x^2 - 12x + 20}{x - 1} \) using the six-step procedure, follow these guidelines. Ensure to draw any asymptotes as dashed lines.
1. **Factor the Numerator and Denominator (if possible):**
- The numerator \( x^2 - 12x + 20 \) can be factored. Finding the roots, we get:
\[
x^2 - 12x + 20 = (x-10)(x-2)
\]
- Therefore, the function can be rewritten as:
\[
f(x) = \frac{(x-10)(x-2)}{x-1}
\]
2. **Find the Vertical Asymptotes:**
- Vertical asymptotes occur where the denominator is zero and the numerator is not zero. Here, the denominator \( x-1 = 0 \) at \( x = 1 \).
- Hence, there is a vertical asymptote at \( x = 1 \).
3. **Find the Horizontal or Oblique (Slant) Asymptotes:**
- To find the horizontal asymptote, compare the degrees of the numerator and the denominator.
- The degree of the numerator is 2, and the degree of the denominator is 1. Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
- To find the oblique asymptote, perform polynomial long division:
\[
\frac{x^2 - 12x + 20}{x - 1} \Rightarrow x - 11 \quad (\text{Quotient}) + \frac{9}{x-1} \quad (\text{Remainder})
\]
- The oblique asymptote is \( y = x - 11 \).
4. **Find the Intercepts:**
- **X-Intercepts:** Set the numerator equal to zero and solve for \( x \):
\[
(x-10)(x-2) = 0 \Rightarrow x = 10, \quad x = 2
\]
- **Y-Intercept:** Set \( x = 0 \):](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3ae0660d-5301-4b30-8361-0f1bb7fa4959%2Fe589a14d-b7af-4612-a219-d234ad151cc5%2Fppj3jml_processed.png&w=3840&q=75)
Transcribed Image Text:### Graphing a Rational Function Using the Six-Step Procedure
To graph the rational function \( f(x) = \frac{x^2 - 12x + 20}{x - 1} \) using the six-step procedure, follow these guidelines. Ensure to draw any asymptotes as dashed lines.
1. **Factor the Numerator and Denominator (if possible):**
- The numerator \( x^2 - 12x + 20 \) can be factored. Finding the roots, we get:
\[
x^2 - 12x + 20 = (x-10)(x-2)
\]
- Therefore, the function can be rewritten as:
\[
f(x) = \frac{(x-10)(x-2)}{x-1}
\]
2. **Find the Vertical Asymptotes:**
- Vertical asymptotes occur where the denominator is zero and the numerator is not zero. Here, the denominator \( x-1 = 0 \) at \( x = 1 \).
- Hence, there is a vertical asymptote at \( x = 1 \).
3. **Find the Horizontal or Oblique (Slant) Asymptotes:**
- To find the horizontal asymptote, compare the degrees of the numerator and the denominator.
- The degree of the numerator is 2, and the degree of the denominator is 1. Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
- To find the oblique asymptote, perform polynomial long division:
\[
\frac{x^2 - 12x + 20}{x - 1} \Rightarrow x - 11 \quad (\text{Quotient}) + \frac{9}{x-1} \quad (\text{Remainder})
\]
- The oblique asymptote is \( y = x - 11 \).
4. **Find the Intercepts:**
- **X-Intercepts:** Set the numerator equal to zero and solve for \( x \):
\[
(x-10)(x-2) = 0 \Rightarrow x = 10, \quad x = 2
\]
- **Y-Intercept:** Set \( x = 0 \):
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