College Algebra: Key Concepts, Problem Solving, and Function Analysis

Graphing and Analyzing Functions

Finding X-Intercepts and Zeros

The x-intercepts of a function are the points where the graph crosses the x-axis. The zeros of a function are the x-values for which the function equals zero.

• X-intercepts: Points of the form $(x, 0)$ on the graph.

• Zeros: Solutions to $f(x) = 0$.

• Example: For a graph with x-intercepts at $(-2, 0)$ and $(2, 0)$, the zeros are $x = -2$ and $x = 2$.

Determining Solutions to Equations

To check if a point $(x, y)$ is a solution to an equation, substitute $x$ and $y$ into the equation and verify if the equality holds.

• Example: Is $(3, 1)$ a solution to $y = 2x + 4$?

• Substitute: $1 = 2(3) + 4 = 6 + 4 = 10$ (not true), so $(3, 1)$ is not a solution.

Graphing Linear Equations

Linear equations can be graphed by finding points or using the slope-intercept form $y = mx + b$.

• Example: Graph $y = 3x + 1$ by finding points or using the slope ($m = 3$) and y-intercept ($b = 1$).

Finding the Slope Between Two Points

The slope $m$ of a line through points $(x_1, y_1)$ and $(x_2, y_2)$ is:

• $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Example: For $(3, 6)$ and $(4, 1)$, $m = \frac{1 - 6}{4 - 3} = \frac{-5}{1} = -5$.

Solving Equations and Inequalities

Solving Quadratic Equations

Quadratic equations of the form $ax^2 + bx + c = 0$ can be solved using the quadratic formula:

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

• Example: $x^2 + 5x + 6 = 0$ gives $x = \frac{-5 \pm \sqrt{25 - 24}}{2} = \frac{-5 \pm 1}{2}$, so $x = -2, -3$.

Solving Linear Inequalities

To solve inequalities, isolate the variable and express the solution as an interval.

• Example: $-5 < x + 2 < 10$

• Subtract 2: $-7 < x < 8$

• Interval notation: $(-7, 8)$

Solving Absolute Value and Radical Equations

• Example: $\sqrt{1 - x} = 9$

• Square both sides: $1 - x = 81$

• Solve: $x = 1 - 81 = -80$

Piecewise Functions and Function Operations

Evaluating Piecewise Functions

A piecewise function is defined by different expressions depending on the input value.

• Example: $f(x) = \begin{cases} x + 1 & x \leq 2 \\ 3 & x > 2 \end{cases}$

• Evaluate $f(-6) = -6 + 1 = -5$; $f(2) = 2 + 1 = 3$; $f(5) = 3$; $f(7) = 3$.

Finding Zeros of Functions

• Zero: The value of $x$ for which $f(x) = 0$.

• Example: For $f(x) = 7 - x$, set $0 = 7 - x \implies x = 7$.

Function Operations

Given functions $f(x)$ and $g(x)$, operations like $f - g$ can be performed by subtracting their outputs.

• Example: If $f(x) = x^2 + 1$, $g(x) = 7 - x$, then $f - g = x^2 + 1 - (7 - x) = x^2 + x - 6$.

• Evaluate at $x = 3$: $f - g(3) = 3^2 + 3 - 6 = 9 + 3 - 6 = 6$.

Graphing and Intervals

Graphing Solutions to Inequalities

After solving an inequality, represent the solution on a number line and in interval notation.

• Example: $2 \leq x$ is $[2, \infty)$ on the number line.

Functions and Their Properties

Vertical and Horizontal Line Tests

The vertical line test determines if a graph represents a function. The horizontal line test checks if a function is one-to-one (injective).

• If any vertical line crosses the graph more than once, it is not a function.

• If any horizontal line crosses the graph more than once, the function is not one-to-one.

• Example: A circle fails the vertical line test (not a function). A parabola passes the vertical but fails the horizontal line test (function, but not one-to-one).

Complex Numbers and Operations

Imaginary Numbers and Powers of i

The imaginary unit $i$ is defined as $i = \sqrt{-1}$. Powers of $i$ repeat every four terms.

• $i^2 = -1$

• $i^3 = -i$

• $i^4 = 1$

• Example: $i^{35} = i^{(4 \times 8) + 3} = i^3 = -i$

Adding Complex Numbers

To add complex numbers, add real parts and imaginary parts separately.

• Example: $(2 + 2i) + (7 + 4i) = (2 + 7) + (2i + 4i) = 9 + 6i$

Parallel and Perpendicular Lines

Criteria for Parallelism and Perpendicularity

• Parallel lines: Have the same slope ($m_1 = m_2$).

• Perpendicular lines: The product of their slopes is $-1$ ($m_1 \cdot m_2 = -1$).

• Example: If one line has slope $\frac{2}{3}$, a perpendicular line has slope $-\frac{3}{2}$.

Even and Odd Functions

Definitions and Examples

• Odd function: Satisfies $f(-x) = -f(x)$; symmetric about the origin. Example: $f(x) = 3x^3 + 4x + 2x$ (all odd powers).

• Even function: Satisfies $f(-x) = f(x)$; symmetric about the y-axis. Example: $f(x) = -2x^4 + 5x^2 + 1$ (all even powers).