Graphs of Rational, Exponential, and Logarithmic Functions: Key Concepts and Applications

CHAPTER 3

Section 3.5: Graphs of Rational Functions

Rational functions are quotients of polynomials and their graphs exhibit unique features such as holes, vertical and horizontal asymptotes. Understanding these features is essential for analyzing and sketching rational function graphs.

• x- and y-intercepts: The points where the graph crosses the x-axis and y-axis, found by setting $y=0$ and $x=0$ respectively.

• Holes vs. Vertical Asymptotes: A hole occurs at a value of $x$ where both the numerator and denominator are zero (common factor cancels). A vertical asymptote occurs where the denominator is zero but the numerator is not.

• Horizontal Asymptotes: Determined by comparing the degrees of the numerator and denominator polynomials:

  • If degree numerator < degree denominator: $y=0$ is the horizontal asymptote.

  • If degrees are equal: $y=\frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$

  • If degree numerator > degree denominator: No horizontal asymptote (may have an oblique/slant asymptote).

• Graphing Rational Functions: Use the 7-step process:

  1. Find domain.

  2. Find intercepts.

  3. Find holes.

  4. Find vertical asymptotes.

  5. Find horizontal/oblique asymptotes.

  6. Plot additional points.

  7. Sketch the graph.

• Proper Notation for Asymptotes: Use equations such as $y=a$, $x=a$, or $x=n\pi$ for vertical/horizontal asymptotes.

Example: For $f(x) = \frac{x^2-1}{x-1}$, there is a hole at $x=1$ and a slant asymptote at $y=x+1$.

CHAPTER 4

Section 4.1: Intro to Exponential Functions and Graphs

Exponential functions have the form $y = b^x$ ($b > 0$, $b \neq 1$) and are fundamental in modeling growth and decay. Their graphs have distinct characteristics and can be transformed by shifting or reflecting.

• Basic Shape: The graph of $y = b^x$ is increasing if $b > 1$ and decreasing if $0 < b < 1$.

• Transformations: Shifting and reflecting the graph changes its position and orientation. For example, $y = b^{x-h} + k$ shifts right by $h$ and up by $k$.

• Domain and Range: The domain is $(-\infty, \infty)$; the range is $(0, \infty)$ for the basic function.

• Compound Interest: Monthly compounding uses $A = P\left(1+\frac{r}{n}\right)^{nt}$; continuous compounding uses $A = Pe^{rt}$.

Example: $y = 2^x$ has a horizontal asymptote at $y=0$ and passes through $(0,1)$.

Section 4.2: Introduction to Logarithms: Definitions and Graphs

Logarithms are the inverses of exponential functions. The logarithmic function $y = \log_b x$ answers the question: "To what power must $b$ be raised to get $x$?"

• Definition: $y = \log_b x \iff b^y = x$

• Common Logarithm: Base 10, written as $\log x$

• Natural Logarithm: Base $e$, written as $\ln x$

• Simplifying Logarithmic Expressions: Use properties such as product, quotient, and power rules:

  • Product: $\log_b(xy) = \log_b x + \log_b y$

  • Quotient: $\log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y$

  • Power: $\log_b(x^r) = r\log_b x$

• Domain of Logarithmic Functions: $x > 0$

• Graphing Logarithmic Functions: The graph of $y = \log_b x$ has a vertical asymptote at $x=0$ and passes through $(1,0)$.

• Relationship to Exponential Functions: The graphs are symmetric about $y=x$.

• Transformations: Shifting and reflecting the graph changes its position and orientation.

Example: $y = \ln x$ has a vertical asymptote at $x=0$ and passes through $(1,0)$.

Section 4.3: Properties of Logs

Logarithmic properties allow for the simplification and expansion of expressions, which is essential for solving equations and analyzing functions.

• Simplifying Expressions: Use cancellation and combination properties.

• Expand/Condense: Apply product, quotient, and power rules to rewrite logarithmic expressions.

• Change-of-Base Theorem: $\log_b x = \frac{\log_a x}{\log_a b}$, useful for evaluating logs with different bases.

• Calculator Approximation: Use the change-of-base formula to approximate logarithms in terms of common or natural logs.

Example: $\log_2 8 = \frac{\log_{10} 8}{\log_{10} 2} = 3$

Section 4.4: Solving Exponential and Logarithmic Equations

Equations involving exponentials and logarithms can be solved using their properties and definitions. This is crucial for applications in science, finance, and engineering.

• Exponential Equations: Set equal bases and solve for the exponent, e.g., $2^x = 8 \implies x=3$.

• Exponential Equations with Unlike Bases: Take logarithms of both sides, e.g., $3^x = 7 \implies x = \frac{\ln 7}{\ln 3}$.

• Logarithmic Equations: Use the definition of logarithms to solve for the variable, e.g., $\log_b x = y \implies x = b^y$.

Example: Solve $\ln x = 2 \implies x = e^2$.

Section 4.5: Applications of Exponential and Logarithmic Equations

Exponential and logarithmic equations are widely used in real-world applications such as population growth, radioactive decay, and financial modeling.

• Population Growth: $P = P_0 e^{rt}$

• Radioactive Decay: $N = N_0 e^{-kt}$

• Compound Interest: $A = Pe^{rt}$

• Solving for Time or Rate: Use logarithms to isolate variables in exponential models.

Example: If $A = 1000 e^{0.05t}$, solve for $t$ when $A = 2000$: $2000 = 1000 e^{0.05t} \implies t = \frac{\ln 2}{0.05}$

Property	Exponential Functions	Logarithmic Functions
Domain	$(-\infty, \infty)$	$(0, \infty)$
Range	$(0, \infty)$	$(-\infty, \infty)$
Asymptote	Horizontal ($y=0$)	Vertical ($x=0$)
Intercept	$(0,1)$	$(1,0)$