Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transformation of Graphs
The word ‘transformation’ means modification. Transformation of the graph of a function is a process by which we modify or change the original graph and make a new graph.
Exponential Functions
The exponential function is a type of mathematical function which is used in real-world contexts. It helps to find out the exponential decay model or exponential growth model, in mathematical models. In this topic, we will understand descriptive rules, concepts, structures, graphs, interpreter series, work formulas, and examples of functions involving exponents.
Question
F: #3.
![**Graphing Rational Functions with Asymptotes and Intercepts**
Consider the rational function:
\[ r(x) = \frac{2x^2 + 7x - 4}{x^2 + x - 2} \]
**Steps to Graph the Rational Function, Identify Asymptotes and Intercepts**
1. **Factor the Numerator and Denominator:**
- Find the factors of the numerator \(2x^2 + 7x - 4\).
- Find the factors of the denominator \(x^2 + x - 2\).
2. **Determine the Asymptotes:**
- **Vertical Asymptotes:** Set the denominator equal to zero and solve for \(x\). These values are where the function may have vertical asymptotes.
- **Horizontal Asymptotes:** Consider the degrees of the numerator and denominator polynomials:
- If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\).
- If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.
- If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote (but there may be an oblique asymptote).
3. **Find the Intercepts:**
- **x-intercepts:** Set the numerator equal to zero and solve for \(x\). These are the \(x\)-values where the function crosses the \(x\)-axis.
- **y-intercept:** Evaluate the function at \(x = 0\) to find the point where the function crosses the \(y\)-axis.
4. **Graph the Function:**
- Plot the identified intercepts and asymptotes on the graph.
- Sketch the general shape of the graph around these key points ensuring that the function approaches the asymptotes appropriately.
In practice, this function would first be factored as follows:
\[2x^2 + 7x - 4 = (2x - 1)(x + 4)\]
\[x^2 + x - 2 = (x + 2)(x - 1)\]
From here, the asymptotes and intercepts identified through solving equations will guide the graphing process.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0c34de7e-5935-4303-bb64-ce025f7422a2%2Fd68b5ab1-71cb-42db-a0c0-3245645e4eeb%2Fub2k6ob_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Graphing Rational Functions with Asymptotes and Intercepts**
Consider the rational function:
\[ r(x) = \frac{2x^2 + 7x - 4}{x^2 + x - 2} \]
**Steps to Graph the Rational Function, Identify Asymptotes and Intercepts**
1. **Factor the Numerator and Denominator:**
- Find the factors of the numerator \(2x^2 + 7x - 4\).
- Find the factors of the denominator \(x^2 + x - 2\).
2. **Determine the Asymptotes:**
- **Vertical Asymptotes:** Set the denominator equal to zero and solve for \(x\). These values are where the function may have vertical asymptotes.
- **Horizontal Asymptotes:** Consider the degrees of the numerator and denominator polynomials:
- If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\).
- If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.
- If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote (but there may be an oblique asymptote).
3. **Find the Intercepts:**
- **x-intercepts:** Set the numerator equal to zero and solve for \(x\). These are the \(x\)-values where the function crosses the \(x\)-axis.
- **y-intercept:** Evaluate the function at \(x = 0\) to find the point where the function crosses the \(y\)-axis.
4. **Graph the Function:**
- Plot the identified intercepts and asymptotes on the graph.
- Sketch the general shape of the graph around these key points ensuring that the function approaches the asymptotes appropriately.
In practice, this function would first be factored as follows:
\[2x^2 + 7x - 4 = (2x - 1)(x + 4)\]
\[x^2 + x - 2 = (x + 2)(x - 1)\]
From here, the asymptotes and intercepts identified through solving equations will guide the graphing process.
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