College Algebra Study Notes: Rational Functions, Polynomials, Zeros, and Logarithms

Rational Functions and Asymptotes

Vertical, Horizontal, and Oblique Asymptotes

Rational functions are quotients of polynomials and often have asymptotes, which are lines the graph approaches but never touches. Asymptotes can be vertical, horizontal, or oblique (slant).

• Vertical Asymptotes (VA): Occur where the denominator is zero and the numerator is nonzero. For $f(x) = \frac{x^2 + 16}{x^2 - 25}$, set $x^2 - 25 = 0$ to find $x = 5$ and $x = -5$.

• Horizontal Asymptotes (HA): Determined by comparing degrees of numerator and denominator. If degrees are equal, HA is $y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$. For $f(x) = \frac{x^2 + 16}{x^2 - 25}$, HA is $y = 1$.

• Oblique Asymptotes: Occur when the degree of the numerator is exactly one more than the denominator. Divide numerator by denominator to find the equation.

• Domain: All real numbers except where the denominator is zero.

Example: For $f(x) = \frac{x^2 + 16}{x^2 - 25}$, VA: $x = 5$, $x = -5$; HA: $y = 1$; Domain: $(-\infty, -5) \cup (-5, 5) \cup (5, \infty)$.

Polynomial Division and Long Division

Dividing Polynomials

Long division is used to divide polynomials, similar to numerical long division. The result is a quotient and possibly a remainder.

• Steps:

  1. Divide the leading term of the dividend by the leading term of the divisor.

  2. Multiply the divisor by this result and subtract from the dividend.

  3. Repeat with the new polynomial.

• Example: Divide $5x^4 - 13x^3 - 13x^2 + 22x - 2$ by $x - 3$.

Additional info: The quotient will be a cubic polynomial plus a remainder over $x-3$.

Polynomial Functions: End Behavior

Leading Coefficient Test

The end behavior of a polynomial function is determined by its degree and leading coefficient.

• Odd Degree: If leading coefficient is positive, graph falls left and rises right. If negative, rises left and falls right.

• Even Degree: Both ends rise (positive) or fall (negative).

• Example: For $f(x) = 19x^3 - 9x^2 + 2x - 8$, degree is 3 (odd), leading coefficient is 19 (positive): falls left, rises right.

Quadratic Functions: Zeros and Vertex

Finding Zeros Given Vertex

A quadratic function can be written in vertex form: $f(x) = a(x-h)^2 + k$, where $(h, k)$ is the vertex. If one zero is known, the other can be found using symmetry.

• Example: Vertex at $(8, 4)$, one zero at $x = 4$. The other zero is at $x = 12$ (since vertex is midpoint between zeros).

Descartes' Rule of Signs

Determining Number of Real Zeros

Descartes' Rule of Signs helps predict the number of positive and negative real zeros of a polynomial by counting sign changes in $f(x)$ and $f(-x)$.

• Positive Real Zeros: Count sign changes in $f(x)$.

• Negative Real Zeros: Count sign changes in $f(-x)$.

• Example: For $f(x) = 6x^4 - x^3 - 4x^2 - 2x + 9$, there are 2 or 0 positive real zeros, and 2 or 0 negative real zeros.

Intermediate Value Theorem

Existence of Real Zeros

The Intermediate Value Theorem states that if a continuous function changes sign over an interval, there is at least one real zero in that interval.

• Example: For $f(x) = -4x^3 + 9x^2 + 2x - 1$, if $f(0) = -1 < 0$ and $f(2) = 7 > 0$, there is a zero between 0 and 2.

Rational Functions: Graphs and Equations

Identifying Rational Functions from Graphs

Given a graph, the equation of a rational function can be deduced by analyzing asymptotes, intercepts, and behavior.

• Example: If the graph has a vertical asymptote at $x = 5$ and a zero at $x = 4$, the function may be $f(x) = \frac{5(x-4)}{x-5}$.

Exponential and Logarithmic Functions

Exponential and Logarithmic Equations

Logarithmic equations can be rewritten in exponential form and vice versa. The properties of logarithms allow expansion and simplification of expressions.

• Exponential Form: $\log_b a = c$ is equivalent to $b^c = a$.

• Example: $\log_7 343 = 6$ is equivalent to $7^6 = 343$.

• Solving Exponential Equations: $x^{3/2} = 4913$; solve by raising both sides to the reciprocal power: $x = 4913^{2/3}$.

Transformations of Exponential Functions

Transformations shift, stretch, or compress the graph of exponential functions. The domain and range are affected accordingly.

• Example: $g(x) = e^{5x}$ has domain $(-\infty, \infty)$ and range $(0, \infty)$; horizontal asymptote $y = 0$.

Solving Logarithmic Equations

To solve equations involving logarithms, use properties such as the product, quotient, and power rules.

• Example: $\log_4(x^2 - 225) - \log_4(x + 15) = 1$ can be rewritten as $\log_4\left(\frac{x^2 - 225}{x + 15}\right) = 1$, so $\frac{x^2 - 225}{x + 15} = 4^1 = 4$.

Expanding Logarithmic Expressions

Logarithmic expressions can be expanded using the properties:

• 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: $\log_b(x^r) = r \log_b x$

• Example: $\log\left(1000\sqrt{2-y}/(3y+8^3)\right)$ expands to $2 + 4\log y + \frac{1}{2}\log(2-y) - \log 3 - 3\log(y+8)$.