College Algebra Study Notes: Circles, Functions, Quadratics, Polynomials, and Rational Functions

Equations of Circles

Standard Form of a Circle

The equation of a circle in the coordinate plane is typically written in standard form, which allows us to easily identify the center and radius.

• Standard Form: $(x - h)^2 + (y - k)^2 = r^2$

• Center: $(h, k)$

• Radius: $r$

Example: The equation $(x + 1)^2 + (y - 4)^2 = 9$ represents a circle with center $(-1, 4)$ and radius $3$. Application: To graph a circle, plot the center and use the radius to mark points in all directions. Additional info: The general form $x^2 + y^2 + Dx + Ey + F = 0$ can be converted to standard form by completing the square.

Functions and Function Notation

Evaluating Functions

A function assigns each input exactly one output. Function notation uses $f(x)$ to denote the output for input $x$.

• To evaluate: Substitute the given value into the function.

• Example: If $f(x) = 2x - 3$, then $f(5) = 2(5) - 3 = 7$.

Composite Functions

Composite functions combine two functions, written as $(fog)(x)=f(g(x))$.

• Example: If $f(x) = x^2$ and $g(x) = x + 1$, then $(fog)(2)=f(g(2))=f(3)=9$.

Quadratic Functions and Parabolas

Standard and Vertex Form

Quadratic functions are polynomials of degree 2 and graph as parabolas.

• Standard Form: $y = ax^2 + bx + c$

• Vertex Form: $y = a(x - h)^2 + k$

• Vertex: $(h, k)$

• Axis of Symmetry: $x = h$

Example: For $y = x^2 - 4x + 3$, the vertex is at $x = 2$, $y = -1$. Application: The vertex is the maximum or minimum point of the parabola.

Polynomial Functions

Division and Synthetic Division

Polynomial division is used to divide polynomials by binomials. Synthetic division is a shortcut for dividing by $x - c$.

• Steps: Write coefficients, use the zero of the divisor, and perform synthetic division.

• Example: Divide $x^3 - 4x^2 + 5x - 2$ by $x - 1$ using synthetic division.

Remainder and Factor Theorems

The Remainder Theorem states that the remainder of $f(x)$ divided by $x - c$ is $f(c)$. The Factor Theorem states that $x - c$ is a factor if $f(c) = 0$.

• Example: If $f(2) = 0$, then $x - 2$ is a factor of $f(x)$.

Rational Functions and Asymptotes

Vertical, Horizontal, and Oblique Asymptotes

Rational functions are quotients of polynomials. Asymptotes describe the behavior of the graph at extreme values or where the function is undefined.

• Vertical Asymptotes: Occur at zeros of the denominator (where the function is undefined).

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

• Oblique Asymptotes: Occur when the degree of the numerator is one more than the denominator.

Example: For $f(x) = rac{x^2 - 1}{x - 3}$, vertical asymptote at $x = 3$, oblique asymptote at $y = x + 3$.

Roots and Zeros of Polynomial Functions

Finding Zeros

Zeros of a polynomial are values of $x$ where $f(x) = 0$. They can be found by factoring or using the Rational Root Theorem.

• Rational Root Theorem: Possible rational roots are $rac{ ext{factors of constant}}{ ext{factors of leading coefficient}}$.

• Example: For $f(x) = x^3 - 2x^2 + x - 2$, possible rational roots are $rac{ ext{factors of } -2}{ ext{factors of } 1}$, i.e., $rac{ ext{±1, ±2}}{1}$.

Tables of Values and Graphing

Using Tables to Graph Functions

Tables of values help plot points for graphing functions. Each input $x$ is paired with an output $f(x)$.

x	f(x)
1	3
2	5
3	8

Application: Plot each $(x, f(x))$ pair on the coordinate plane to visualize the function.

Summary Table: Types of Equations and Their Key Properties

Type	Standard Form	Key Properties
Circle	$(x - h)^2 + (y - k)^2 = r^2$	Center $(h, k)$, radius $r$
Quadratic	$y = ax^2 + bx + c$	Vertex, axis of symmetry, parabola shape
Polynomial	$a_nx^n + ext{...} + a_0$	Degree, zeros, end behavior
Rational	$rac{P(x)}{Q(x)}$	Asymptotes, domain restrictions

Additional info: These notes cover foundational topics in College Algebra, including equations of circles, function evaluation, quadratic and polynomial functions, rational functions, and graphing techniques. Each topic is essential for understanding algebraic relationships and graphing in the coordinate plane.