College Algebra Study Guide: Functions, Graphs, and Equations

Lines and Linear Equations

Distance, Slope, and Equation of a Line

Understanding lines in the coordinate plane is fundamental in College Algebra. Key concepts include calculating the distance between points, finding the slope, and writing equations of lines.

• Distance Between Two Points: The distance between points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Slope of a Line: The slope $m$ between two points is: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Equation of a Line: The equation of a line with slope $m$ passing through $(x_1, y_1)$ is: $y - y_1 = m(x - x_1)$

• Parallel and Perpendicular Lines: Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals.

• Example: Find the equation of the line passing through $(2, 3)$ and $(4, 7)$. $m = \frac{7-3}{4-2} = 2$ Equation: $y - 3 = 2(x - 2)$

Intercepts and Circles

Finding Intercepts

Intercepts are points where a graph crosses the axes.

• x-intercept: Set $y = 0$ and solve for $x$.

• y-intercept: Set $x = 0$ and solve for $y$.

• Example: For $4x - 5y = 20$, set $y = 0$ to find $x$-intercept: $x = 5$; set $x = 0$ to find $y$-intercept: $y = -4$.

Equation of a Circle

The standard form of a circle's equation is:

• $(x - h)^2 + (y - k)^2 = r^2$ where $(h, k)$ is the center and $r$ is the radius.

• Example: $x^2 + y^2 - 2x + 4y = 0$ can be rewritten by completing the square.

Graphing Linear Equations

Slope-Intercept Form

The slope-intercept form of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Parallel Lines: Same slope, different y-intercepts.

• Perpendicular Lines: Slopes are negative reciprocals.

• Example: A line parallel to $y = 2x + 3$ and passing through $(4, 1)$: $y - 1 = 2(x - 4)$

Functions and Their Properties

Definition and Domain

A function is a relation where each input has exactly one output. The domain is the set of all possible input values.

• Example: $y = \frac{2x}{x^2 - 5}$ has domain $x \neq \pm \sqrt{5}$

• Vertical Line Test: A graph is a function if no vertical line intersects it more than once.

Evaluating Functions

• To evaluate $f(x)$, substitute the value of $x$ into the function.

• Example: If $f(x) = x^2 - 4x + 1$, then $f(3) = 9 - 12 + 1 = -2$

Domain of Rational and Radical Functions

• Rational Functions: Exclude values that make the denominator zero.

• Radical Functions: For even roots, exclude values that make the radicand negative.

• Example: $y = \sqrt{2x - 5}$ has domain $x \geq 2.5$

Graphing and Transformations

Basic Graphs and Intercepts

Key features of graphs include intercepts, symmetry, and transformations.

• x-intercept: Where $y = 0$

• y-intercept: Where $x = 0$

• Symmetry: Even functions are symmetric about the y-axis; odd functions about the origin.

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

Transformations of Functions

Transformations shift, stretch, or reflect graphs.

• Vertical Shift: $f(x) + c$ shifts up by $c$ units.

• Horizontal Shift: $f(x + c)$ shifts left by $c$ units.

• Reflection: $-f(x)$ reflects over the x-axis.

• Vertical Stretch: $a f(x)$ stretches by factor $a$.

• Example: $y = \sqrt{x}$, then $y = 2\sqrt{x}$ is a vertical stretch by 2; $y = -\sqrt{x}$ is a reflection over the x-axis.

Analyzing Graphs

Local Maxima and Minima

Local maxima and minima are the highest and lowest points in a region of a graph.

• Local Maximum: Point where the function changes from increasing to decreasing.

• Local Minimum: Point where the function changes from decreasing to increasing.

Absolute Value and Piecewise Functions

Absolute value functions create a 'V' shape graph. Piecewise functions are defined by different expressions over different intervals.

• Example: $f(x) = |x|$

Summary Table: Function Transformations

Transformation	Equation	Effect on Graph
Vertical Shift	$f(x) + c$	Up by $c$ units
Horizontal Shift	$f(x + c)$	Left by $c$ units
Reflection over x-axis	$-f(x)$	Flips graph upside down
Vertical Stretch	$a f(x)$	Stretches graph by factor $a$

Additional info:

• Some questions involve sketching graphs and describing transformations, which are essential for visual understanding in algebra.

• Questions about domain, range, and intercepts reinforce foundational skills for analyzing functions.

• Reflecting and stretching functions are common transformations tested in College Algebra.