College Algebra: Functions, Polynomials, and Quadratics Study Guide

Functions and Their Properties

Solving Inequalities Involving Absolute Value

Absolute value inequalities are equations that involve the absolute value function, which measures the distance of a number from zero on the real number line.

• Definition: The absolute value of a number x is denoted |x| and is defined as: $|x| = \begin{cases} x, & x \geq 0 \\ -x, & x < 0 \end{cases}$

• Solving |x + 1| > 3: This splits into two inequalities: $x + 1 > 3$ or $x + 1 < -3$ So, $x > 2$ or $x < -4$.

• Solving |x + 1| < 3: This becomes: $-3 < x + 1 < 3$ So, $-4 < x < 2$.

• Solving |x + 1| \geq 3: This is: $x + 1 \geq 3$ or $x + 1 \leq -3$ So, $x \geq 2$ or $x \leq -4$.

Graphing Functions and Transformations

Understanding how to graph functions and their transformations is essential in algebra. Transformations include shifts, reflections, and stretches/compressions.

• Graph of |f(x)|: To graph $|f(x)|$, reflect any portion of the graph of $f(x)$ that is below the x-axis over the x-axis.

• Piecewise Functions: A function defined by different expressions over different intervals. Example: $f(x) = \begin{cases} 3x + 2, & x < 1 \\ x - 5, & x \geq 2 \end{cases}$

• Domain: The set of all possible input values (x-values) for which the function is defined.

• Range: The set of all possible output values (y-values) of the function.

• Intervals of Continuity: The intervals where the function is continuous (no breaks, jumps, or holes).

Operations with Functions

Function Composition and Evaluation

Functions can be combined using operations such as addition, subtraction, multiplication, division, and composition.

• Composition: $(f \circ g)(x) = f(g(x))$

• Examples:

  • $(f \circ g)(|x|)$: Apply $g$ to $|x|$, then $f$ to the result.

  • $(g|x|)$: Apply $g$ to $|x|$.

  • $(f - g)(x)$: Subtract $g(x)$ from $f(x)$.

  • $(f - g)(1)$: Evaluate $f(1) - g(1)$.

  • $\left(\frac{f}{g}\right)(3)$: Evaluate $\frac{f(3)}{g(3)}$.

Polynomials and Their Properties

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the expression.

• Example: The degree of $2x^4 + 3x^2 - 7$ is 4.

Square Roots and Complex Numbers

Square roots of negative numbers involve the imaginary unit $i$, where $i^2 = -1$.

• Examples:

  • $\sqrt{-4} = 2i$

  • $\sqrt{18} = 3\sqrt{2}$

  • $\sqrt{-28} = 2\sqrt{7}i$

Operations with Complex Numbers

• Multiplication: Use distributive property and $i^2 = -1$. Example: $(3 + i)(4 - i)$

• Division: Multiply numerator and denominator by the conjugate of the denominator. Example: $\frac{2 + i}{3 - 2i}$

Quadratic Functions and Vertex Form

Quadratic functions can be written in standard form $ax^2 + bx + c$ or vertex form $a(x - h)^2 + k$.

• Vertex: The point $(h, k)$ where the parabola reaches its maximum or minimum.

• Finding the Vertex: For $f(x) = ax^2 + bx + c$, vertex is at $x = -\frac{b}{2a}$.

• Rewriting in Vertex Form: Complete the square to rewrite $f(x)$ in vertex form.

End Behavior of Polynomials

The end behavior describes how the function behaves as $x \to \infty$ or $x \to -\infty$.

• Even Degree, Positive Leading Coefficient: Both ends up.

• Even Degree, Negative Leading Coefficient: Both ends down.

• Odd Degree, Positive Leading Coefficient: Left down, right up.

• Odd Degree, Negative Leading Coefficient: Left up, right down.

Solving Quadratic Equations

Factoring and Solving

Quadratic equations can be solved by factoring, completing the square, or using the quadratic formula.

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

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

• Example: $2x^2 + 6x - 12 = 0$

Zeros of a Function

The zeros (roots) of a function are the x-values where $f(x) = 0$.

• Factoring: If $x = c$ is a zero, then $(x - c)$ is a factor.

Polynomial Division

Synthetic and Long Division

Polynomials can be divided using synthetic division (for divisors of the form $x - c$) or long division.

• Synthetic Division: A shortcut method for dividing by $x - c$.

• Example: Divide $x^4 + 4x^4$ by $x + 2$ using synthetic division.

Factoring Polynomials

Factoring is the process of writing a polynomial as a product of its factors.

• Difference of Squares: $a^2 - b^2 = (a - b)(a + b)$

• Example: $x^2 - 64 = (x - 8)(x + 8)$

• Factoring by Grouping: Used for higher-degree polynomials.

Graph Analysis

Intervals of Increase and Decrease

To determine where a function is increasing or decreasing, analyze the slope of the graph.

• Increasing: The function rises as x increases.

• Decreasing: The function falls as x increases.

• Example: For a parabola opening upwards, the function decreases to the vertex and increases after.

Summary Table: Key Polynomial Properties

Property	Description	Example
Degree	Highest exponent of x	$3x^4 + 2x^2$ has degree 4
Leading Coefficient	Coefficient of the highest degree term	In $-2x^3 + x$, leading coefficient is -2
End Behavior	Behavior as $x \to \pm\infty$	Even degree, positive: both ends up
Zeros	Values where $f(x) = 0$	$x^2 - 9 = 0$ has zeros at $x = 3, -3$