College Algebra Study Guide: Polynomial, Rational, Exponential, and Logarithmic Functions (Sections 3.5–4.4)

Polynomial and Rational Functions

Section 3.5: Graphs of Polynomial Functions

Polynomial functions are algebraic expressions involving powers of x with real coefficients. Their graphs exhibit distinct shapes depending on degree and leading coefficient.

• Key Point 1: The degree of the polynomial determines the number of turning points and end behavior.

• Key Point 2: The leading coefficient affects whether the graph rises or falls as x approaches infinity.

• Example: For $f(x) = x^3 - 2x^2 + x$, the graph has two turning points and end behavior matching $x^3$.

Section 3.6: Graphs of Rational Functions

Rational functions are quotients of polynomials. Their graphs may have vertical, horizontal, or oblique asymptotes, and can exhibit discontinuities.

• Key Point 1: Vertical asymptotes occur at zeros of the denominator not canceled by the numerator.

• Key Point 2: Horizontal asymptotes are determined by comparing degrees of numerator and denominator.

• Formula: For $f(x) = \frac{P(x)}{Q(x)}$, vertical asymptotes at $Q(x) = 0$.

• Example: $f(x) = \frac{1}{x-2}$ has a vertical asymptote at $x=2$ and a horizontal asymptote at $y=0$.

Exponential and Logarithmic Functions

Section 4.1: Exponential Functions

Exponential functions have the form $f(x) = a^x$, where $a > 0$ and $a \neq 1$. They model rapid growth or decay.

• Key Point 1: The base $a$ determines growth ($a > 1$) or decay ($0 < a < 1$).

• Key Point 2: The graph passes through $(0,1)$ and is always positive.

• Formula: $f(x) = a^x$

• Example: $f(x) = 2^x$ grows rapidly as $x$ increases.

Section 4.2: Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. The general form is $f(x) = \log_a(x)$, where $a > 0$, $a \neq 1$.

• Key Point 1: The domain is $x > 0$; the range is all real numbers.

• Key Point 2: The graph passes through $(1,0)$ and has a vertical asymptote at $x=0$.

• Formula: $y = \log_a(x) \iff a^y = x$

• Example: $f(x) = \log_{10}(x)$ is the common logarithm.

Section 4.3: Properties of Logarithms

Logarithms have several important properties that simplify expressions and solve equations.

• Key Point 1: Product Rule: $\log_a(xy) = \log_a(x) + \log_a(y)$

• Key Point 2: Quotient Rule: $\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)$

• Key Point 3: Power Rule: $\log_a(x^r) = r\log_a(x)$

• Example: $\log_2(8) = \log_2(2^3) = 3\log_2(2) = 3$

Section 4.4: Exponential and Logarithmic Equations

Equations involving exponentials and logarithms can be solved using properties of these functions.

• Key Point 1: To solve $a^x = b$, take logarithms of both sides: $x = \log_a(b)$

• Key Point 2: To solve $\log_a(x) = b$, rewrite as $x = a^b$

• Example: Solve $2^x = 16$. $x = \log_2(16) = 4$

Function Type	General Form	Key Features
Polynomial	$f(x) = a_nx^n + \ldots + a_0$	Degree, turning points, end behavior
Rational	$f(x) = \frac{P(x)}{Q(x)}$	Asymptotes, discontinuities
Exponential	$f(x) = a^x$	Growth/decay, always positive
Logarithmic	$f(x) = \log_a(x)$	Inverse of exponential, domain $x>0$

Additional info: Section numbers (3.5, 3.6, 4.1, 4.2, 4.3, 4.4) were inferred to correspond to standard College Algebra textbook topics based on context and typical curriculum structure.