College Algebra: Polynomial, Rational, Exponential, Logarithmic Functions, Systems, and Sequences

Polynomial Functions & Graphs

3.1 Polynomial Functions & Graphs

Polynomial functions are a fundamental class of functions in algebra, characterized by their degree and the behavior of their graphs. Understanding their structure and properties is essential for graphing and solving equations.

• Definition: A polynomial function has the form: $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$, where $a_n \neq 0$ and $n$ is a non-negative integer.

• Degree: The highest power of $x$ in the polynomial. Determines:

  • Number of turning points (at most $n-1$)

  • End behavior

• End Behavior: Determined by the leading term $a_nx^n$:

  • Even degree, positive leading coefficient: $\uparrow$ both ends

  • Even degree, negative leading coefficient: $\downarrow$ both ends

  • Odd degree, positive leading coefficient: $\downarrow$ left, $\uparrow$ right

  • Odd degree, negative leading coefficient: $\uparrow$ left, $\downarrow$ right

Skills to Master:

• Identify degree and leading coefficient

• Sketch and describe end behavior

• Find zeros and multiplicity:

  • Even multiplicity: touches x-axis

  • Odd multiplicity: crosses x-axis

Example: $f(x) = x^3 - 2x^2 + x$ is a cubic (degree 3) polynomial with leading coefficient 1. Its end behavior: $\downarrow$ left, $\uparrow$ right.

3.2 Dividing Polynomials

Dividing polynomials is essential for simplifying expressions and finding zeros. Two main methods are used: long division and synthetic division.

• Methods:

  • Long division: Standard algorithm for dividing polynomials.

  • Synthetic division: Shortcut when dividing by $x-c$.

• Remainder Theorem: If you divide $f(x)$ by $x-c$, the remainder is $f(c)$.

• Factor Theorem: $x-c$ is a factor of $f(x)$ if and only if $f(c)=0$.

Example: Divide $f(x) = x^3 - 4x^2 + 5x - 2$ by $x-1$ using synthetic division.

3.3 Real Zeros of Polynomial Functions

Finding the real zeros (roots) of a polynomial is crucial for graphing and solving equations.

• Rational Root Theorem: Possible rational zeros are $\pm$ (factors of constant term)/(factors of leading coefficient).

• Descartes' Rule of Signs:

  • Number of positive real zeros: number of sign changes in $f(x)$.

  • Number of negative real zeros: number of sign changes in $f(-x)$.

• Strategy for Finding Zeros:

  1. List possible rational zeros.

  2. Test using synthetic division.

  3. Factor reduced polynomial.

Example: For $f(x) = 2x^3 - 3x^2 - 8x + 12$, possible rational zeros are $\pm1, \pm2, \pm3, \pm4, \pm6, \pm12$ divided by $\pm1, \pm2$.

3.4 Graphing Polynomial Functions

Graphing polynomials involves analyzing their degree, zeros, and turning points to sketch an accurate graph.

• Identify degree & leading coefficient

• Find zeros & multiplicities

• Find y-intercept

• Determine end behavior

• Plot key points & turning points

Example: Sketch $f(x) = x^2 - 4$ by finding zeros at $x=2$ and $x=-2$, y-intercept at $y=-4$, and end behavior $\uparrow$ both ends.

Rational Functionsytu7

3.5 Rational Functions

Rational functions are quotients of polynomials and exhibit unique features such as asymptotes and holes.

• Definition: $f(x) = \frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$.

• Vertical asymptotes: Zeros of denominator.

• Horizontal asymptotes:

  • Degree numerator < degree denominator: $y=0$

  • Degrees equal: $y=$ ratio of leading coefficients

  • Degree numerator > degree denominator: no horizontal, maybe slant asymptote

• Holes: Common factor in numerator & denominator; hole at that x-value.

Example: $f(x) = \frac{x^2-1}{x-1}$ has a hole at $x=1$.

3.6 Graphing Rational Functions

Graphing rational functions requires identifying asymptotes, holes, and intercepts.

• Factor numerator & denominator

• Identify intercepts

• Find vertical/horizontal/slant asymptotes

• Locate holes

• Sketch behavior near asymptotes

Example: Graph $f(x) = \frac{x+2}{x-1}$: vertical asymptote at $x=1$, horizontal asymptote at $y=1$.

Exponential & Logarithmic Functions

4.1 Exponential Functions

Exponential functions model rapid growth or decay and are defined by a constant base raised to a variable exponent.

• Form: $f(x) = ab^{x}$, where $a \neq 0$, $b > 0$, $b \neq 1$

• Properties:

  • Always positive

  • Domain: all real numbers

  • Range: $(0, \infty)$

  • Growth if $b>1$; decay if $0<b<1$

Example: $f(x) = 2^{x}$ is an exponential growth function.

4.2 Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and are used to solve for exponents.

• Definition: $y = \log_b(x)$ means $b^y = x$, where $b>0$, $b \neq 1$

• Properties:

  • Domain: $(0, \infty)$

  • Range: all real numbers

Example: $\log_{10}(100) = 2$ because $10^2 = 100$.

4.3 Properties of Logarithms

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

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

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

• Power Rule: $\log_b(x^r) = r\log_b(x)$

• Change of Base Formula: $\log_b(a) = \frac{\log_k(a)}{\log_k(b)}$

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

4.4 Exponential & Logarithmic Equations

Solving exponential and logarithmic equations often involves rewriting expressions and applying logarithmic properties.

• Rewrite using log or exp definition

• Isolate exponent or log term

• Convert between forms

• Use properties to simplify

Example: Solve $2^x = 8$ by taking $\log_2$ of both sides: $x = 3$.

Systems of Linear Equations & Matrices

5.1 Solving Linear Systems in Two Variables

Systems of linear equations can be solved using various methods, each with its own advantages.

• Methods:

  • Graphing

  • Substitution

  • Elimination

• Interpretation of Outcomes:

  • One solution (consistent): lines intersect

  • No solution (parallel lines): inconsistent

  • Infinitely many solutions (same line): dependent

• Applications: Break-even, mixture, and rate/time problems

Example: Solve $\begin{cases} x + y = 5 \\ x - y = 1 \end{cases}$ by elimination.

Sequences & Series

7.1 Sequences

A sequence is a function whose domain is the natural numbers. Sequences can be classified as arithmetic or geometric.

• Arithmetic sequence: $a_n = a_1 + (n-1)d$

• Geometric sequence: $a_n = a_1 r^{n-1}$

• Determine:

  • Common difference $d$ (arithmetic)

  • Common ratio $r$ (geometric)

Example: $2, 5, 8, 11, \ldots$ is arithmetic with $d=3$.

7.2 Series

A series is the sum of the terms of a sequence. There are formulas for the sum of arithmetic and geometric series.

• Arithmetic Series: $S_n = \frac{n}{2}(a_1 + a_n)$

• Geometric Series (finite): $S_n = a_1 \frac{1 - r^n}{1 - r}$, $r \neq 1$

• Geometric Series (infinite): $S = \frac{a_1}{1 - r}$, $|r| < 1$

Example: The sum of $1 + 2 + 3 + \ldots + 100$ is $S_{100} = \frac{100}{2}(1 + 100) = 5050$.