Exam 4 Study Guide: Polynomial and Rational Functions, Quadratic Equations, and Related Topics

Polynomial and Rational Functions

Symmetry of Graphs

Understanding the symmetry of a graph is essential for analyzing its behavior and simplifying equations. A graph may be symmetric with respect to the y-axis, x-axis, or origin.

• Y-axis symmetry: If replacing $x$ with $-x$ yields the same equation, the graph is symmetric about the y-axis.

• X-axis symmetry: If replacing $y$ with $-y$ yields the same equation, the graph is symmetric about the x-axis.

• Origin symmetry: If replacing both $x$ with $-x$ and $y$ with $-y$ yields the same equation, the graph is symmetric about the origin.

• Example: The graph of $y = x^2$ is symmetric about the y-axis.

Even, Odd, and Neither Functions

Functions can be classified as even, odd, or neither based on their symmetry properties.

• Even function: $f(-x) = f(x)$ for all $x$ in the domain.

• Odd function: $f(-x) = -f(x)$ for all $x$ in the domain.

• Neither: If neither condition holds, the function is neither even nor odd.

• Example: $f(x) = x^3$ is odd; $f(x) = x^2$ is even.

Transformations of Functions

Transformations alter the appearance of a graph. Common transformations include translations, reflections, stretches, and compressions.

• Translation: Shifts the graph horizontally or vertically.

• Reflection: Flips the graph over a line (e.g., x-axis or y-axis).

• Stretch/Compression: Changes the graph's width or height.

• Example: $f(x) = (x - 2)^2 + 3$ is a translation of $y = x^2$ right by 2 units and up by 3 units.

Polynomial Functions and Their Graphs

Polynomial functions are expressions of the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + ext{...} + a_0$. Their graphs have distinct characteristics based on degree and leading coefficient.

• End behavior: Determined by the degree and sign of the leading coefficient.

• Turning points: A polynomial of degree $n$ has at most $n-1$ turning points.

• Example: $f(x) = x^3 - 2x^2 + x$ is a cubic polynomial.

Factoring and Simplifying Expressions

Factoring is the process of expressing a polynomial as a product of its factors. Simplifying involves combining like terms and reducing expressions.

• Common factoring methods: Factoring out the greatest common factor, grouping, difference of squares, and quadratic factoring.

• Example: $x^2 - 9 = (x - 3)(x + 3)$

Operations with Complex Numbers

Complex numbers are of the form $a + bi$, where $i$ is the imaginary unit ($i^2 = -1$). Operations include addition, subtraction, multiplication, and division.

• Addition/Subtraction: Combine real and imaginary parts separately.

• Multiplication: Use distributive property and $i^2 = -1$.

• Division: Multiply numerator and denominator by the conjugate of the denominator.

• Example: $(2 + 3i) + (1 - 4i) = 3 - i$

Quadratic Equations and Applications

Solving Quadratic Equations

Quadratic equations are of the form $ax^2 + bx + c = 0$. They can be solved by factoring, completing the square, or using the Quadratic Formula.

• Factoring: Express as a product of binomials and set each factor to zero.

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Completing the Square: Rearrange and add terms to form a perfect square trinomial.

• Example: Solve $x^2 - 5x + 6 = 0$ by factoring: $(x - 2)(x - 3) = 0$ so $x = 2$ or $x = 3$.

Analyzing Solutions and Discriminant

The discriminant $D = b^2 - 4ac$ determines the nature of the solutions to a quadratic equation.

• If $D > 0$: Two distinct real solutions.

• If $D = 0$: One real solution (a repeated root).

• If $D < 0$: Two complex solutions.

• Example: For $x^2 + 4x + 5 = 0$, $D = 16 - 20 = -4$ (complex solutions).

Word Problems and Applications

Quadratic equations are used to solve real-world problems involving area, motion, and optimization.

• Set up equations: Translate the problem into a quadratic equation.

• Interpret solutions: Consider the context to determine which solutions are meaningful.

• Example: The height $h$ of a ball thrown upward: $h = -16t^2 + vt + h_0$

Rational Functions and Equations

Solving Rational Equations

Rational equations involve fractions with polynomials in the numerator and denominator. Solutions require finding a common denominator and checking for extraneous solutions.

• Find common denominator: Multiply both sides by the least common denominator (LCD).

• Solve resulting equation: Simplify and solve for the variable.

• Check for extraneous solutions: Substitute solutions back to ensure they do not make any denominator zero.

• Example: $\frac{1}{x} + \frac{2}{x+1} = 3$

Analyzing Rational Functions

Rational functions are of the form $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials. Their graphs have unique features such as asymptotes and holes.

• Vertical asymptotes: Occur where $Q(x) = 0$ and $P(x) \neq 0$.

• Horizontal asymptotes: Determined by the degrees of $P(x)$ and $Q(x)$.

• Holes: Occur where both $P(x)$ and $Q(x)$ are zero for the same $x$ value.

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

Graphing Rational Functions

Graphing rational functions involves identifying intercepts, asymptotes, and behavior near undefined points.

• Find x- and y-intercepts: Set $y = 0$ and $x = 0$ respectively.

• Plot asymptotes: Draw vertical and horizontal asymptotes.

• Analyze end behavior: Consider limits as $x \to \infty$ or $x \to -\infty$.

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

Summary Table: Problem Types and Methods

Problem Type	Method	Example
Symmetry of Graphs	Replace variables, check equation	$y = x^2$ (y-axis symmetry)
Even/Odd Functions	Test $f(-x)$	$f(x) = x^3$ (odd)
Transformations	Apply shifts, reflections, stretches	$f(x) = (x-2)^2 + 3$
Factoring	Find factors, set to zero	$x^2 - 9 = (x-3)(x+3)$
Quadratic Equations	Factoring, Quadratic Formula	$x^2 - 5x + 6 = 0$
Rational Equations	Find LCD, solve, check solutions	$\frac{1}{x} + \frac{2}{x+1} = 3$

Additional info: These objectives cover topics from College Algebra chapters on polynomial and rational functions, quadratic equations, and related graphing and problem-solving skills. The study guide is based on the objectives for Exam 4, sections 7.4–7.5.