BackCollege Algebra Study Guide: Functions, Equations, and Graphs
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Functions and Their Properties
Finding X-Intercepts and Zeros of a Function
The x-intercepts of a function are the points where the graph crosses the x-axis, corresponding to the zeros of the function. The zero of a function is a value of x for which f(x) = 0.
X-intercepts: Points (x, 0) where the graph meets the x-axis.
Zeros: Values of x such that 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 is a solution to an equation, substitute the values into the equation and verify if the equality holds.
Example: Is (3, 1) a solution to y = 2x + 4?
Substitute: 1 = 2(3) + 4 → 1 = 6 + 4 → 1 = 10 (not true)
Conclusion: (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
Find points: If x = 0, y = 1; if x = -1/3, y = 0
Plot these points and draw the line.
Finding the Slope Between Two Points
The slope of a line through points and is given by:
Example: For points (3, 6) and (4, 1):
Solving Equations and Inequalities
Quadratic Equations
Quadratic equations of the form can be solved using the quadratic formula:
Example:
Discriminant:
Solutions:
Solving Linear Inequalities
To solve inequalities, isolate the variable and express the solution as an interval.
Example:
Subtract 2:
Interval notation:
Piecewise Functions and Evaluations
Evaluating Piecewise Functions
A piecewise function is defined by different expressions depending on the input value.
Example:
Evaluate :
Evaluate :
Evaluate and : $3x > 2$)
Finding Zeros of Linear Functions
To find the zero of , set and solve for x.
Solving Radical Equations
To solve equations involving radicals, isolate the radical and square both sides.
Example:
Square both sides:
Function Operations
Subtracting Functions
Given and , .
Example: ,
Evaluate at :
Graphing and Interval Notation
Solving and Graphing Inequalities
After solving an inequality, represent the solution on a number line and in interval notation.
Example:
Add to both sides:
Add 2:
Divide by 6:
Interval notation:
Function Identification and Tests
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.
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
Operations with Complex Numbers
Complex numbers are of the form , where .
Example: (using powers of )
Addition:
Solving Proportions
Solving for a Variable in a Proportion
To solve , cross-multiply and solve for x.
Parallel and Perpendicular Lines
Criteria for Parallel and Perpendicular Lines
Lines are parallel if they have the same slope. Lines are perpendicular if the product of their slopes is -1.
Parallel:
Perpendicular:
Example: Slopes and are perpendicular.
Even and Odd Functions
Definitions and Examples
An odd function satisfies and is symmetric about the origin. An even function satisfies and is symmetric about the y-axis.
Odd function example: (all odd powers)
Even function example: (all even powers)
