College Algebra: Functions, Lines, and Graphs Study Guide

Linear Equations and Forms

Point-Slope, Slope-Intercept, and General Forms

Linear equations can be written in several standard forms, each useful for different purposes in algebra and graphing.

• Point-Slope Form: $y - y_1 = m(x - x_1)$ Used when you know the slope m and a point $(x_1, y_1)$ on the line.

• Slope-Intercept Form: $y = mx + b$ Here, m is the slope and b is the y-intercept.

• General Form: $Ax + By + C = 0$ A more general representation, where A, B, and C are constants.

• Horizontal Line: $y = b$ or $f(x) = b$ All points have the same y-value.

• Vertical Line: $x = a$ All points have the same x-value.

Distance and Midpoint Formulas

These formulas are essential for analyzing points and segments in the coordinate plane.

• Distance Formula: $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ Calculates the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$.

• Midpoint Formula: $\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$ Finds the midpoint of the segment connecting two points.

Equation of a Circle

The standard form for the equation of a circle is:

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

Functions and Their Properties

Definition and Identification of Functions

A function is a relation in which each input (x-value) has exactly one output (y-value).

• Vertical Line Test: A graph represents a function if no vertical line intersects the graph at more than one point.

• Examples:

  • Set of ordered pairs: If any x-value repeats with a different y-value, it is not a function.

  • Equations: If for every x there is only one y, it is a function.

Domain and Range

The domain of a function is the set of all possible input values (x-values), and the range is the set of all possible output values (y-values).

• Finding Domain: Consider restrictions such as division by zero or even roots of negative numbers.

• Finding Range: Analyze the outputs produced by the function for all values in the domain.

Intercepts

• x-intercept(s): Points where the graph crosses the x-axis ($y = 0$).

• y-intercept(s): Points where the graph crosses the y-axis ($x = 0$).

Evaluating Functions

To evaluate a function, substitute the given value for x and simplify.

• Piecewise Functions: Functions defined by different expressions over different intervals of the domain.

• Example: $f(x) = \begin{cases} x - 1 & x < 0 \\ 2x + 3 & x \geq 0 \end{cases}$

Graphing and Transformations

Graphing Functions

Graphing involves plotting points and understanding the shape and key features of the function.

• Quadratic Function: $f(x) = x^2$ is a parabola opening upwards.

• Transformations: Include translations (shifts), reflections, stretches, and compressions.

• Example: $g(x) = -2(x+2)^2 + 1$ is a vertical stretch by 2, reflection over the x-axis, left shift by 2, and up shift by 1.

Graphing Circles

To graph a circle given by $(x - h)^2 + (y - k)^2 = r^2$, plot the center at $(h, k)$ and use the radius $r$ to mark points around the center.

Inverse Functions

Definition and Finding Inverses

The inverse of a function reverses the roles of x and y. If $f(x)$ maps $a$ to $b$, then $f^{-1}(x)$ maps $b$ to $a$.

• Finding the Inverse: Replace $f(x)$ with $y$, swap x and y, and solve for y.

• Example: For $f(x) = 2x + 3$, the inverse is $f^{-1}(x) = \frac{x - 3}{2}$.

• Checking Inverses: $f(g(x)) = x$ and $g(f(x)) = x$ for all x in the domain.

Key Formulas and Examples

• Slope Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Midpoint Formula: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

Sample Table: Forms of Linear Equations

Form	Equation	Key Feature
Point-Slope	$y - y_1 = m(x - x_1)$	Given a point and slope
Slope-Intercept	$y = mx + b$	Given slope and y-intercept
General	$Ax + By + C = 0$	Standard linear form

Applications and Practice Problems

• Find the slope of a line through two points using the slope formula.

• Write equations of lines in different forms given points and slopes.

• Determine if a relation is a function using the vertical line test or by checking for repeated x-values.

• Find the domain and range of a function from its graph.

• Evaluate piecewise functions at given values.

• Find the inverse of a function algebraically.

• Use the distance and midpoint formulas for points in the plane.

• Graph circles and identify their centers and radii.

Additional info: The above notes are based on a set of College Algebra practice questions and reference formulas, covering core topics such as linear equations, functions, graphing, and inverses. The content is structured to provide both conceptual understanding and practical skills for exam preparation.