Algebra 1 Flashcards

Relations and Functions

• A relation pairs inputs with outputs; represented as ordered pairs (x, y).
• A function is a specific type of relation where each input corresponds to exactly one output.
• Different inputs may have the same output in a function, but any input must always map to one output.

Graphing Functions

• The vertical line test determines if a graph represents a function. If any vertical line intersects the graph at more than one point, it is not a function.
• The domain of a function includes all possible input values.
• The range of a function includes all possible output values.

Linear Functions and Equations

• A linear equation can be expressed as y = mx + b, where m (slope) and b (y-intercept) are constants. Its graph is a straight line.
• A linear function is an equation in two variables that graphs to a non-vertical line and maintains a constant rate of change.
• The equation y = 6 is a linear function because it can be rewritten as y = 0x + 6. Conversely, y = x^2 + 6 is not linear as it cannot be expressed in linear form.

Characteristics and Types of Functions

• True/False: All linear functions are linear equations—True; not all linear equations are functions, particularly those graphing vertical lines.
• Characteristics of a linear function include:
  • A single output for each input.
  • Consistency: the same input gives the same output.
  • A line graph that does not intersect itself at more than one point.
  • Representing a constant rate of change.

Domain Types

• Discrete domain consists of specific input values, while a continuous domain includes all numbers within an interval.
• Understanding whether a domain is discrete or continuous often requires context beyond the equation.

Intercepts and Slope

• The x-intercept is where the graph crosses the x-axis, identified by its x-coordinate.
• To identify a linear equation's solution set, two points are required due to the constant rate of change.
• The slope of a non-vertical line is calculated as "rise over run."

Function Forms

• A constant function is visually a horizontal line and can be represented as y = b.
• Slope-intercept form, y = mx + b, specifies m as the slope and b as the y-intercept, which represents the output when x = 0.
• Point-slope form is written as y2 - y1 = m(x2 - x1).

Line Relationships

• Non-vertical lines are parallel if they have the same slope. Vertical lines are inherently parallel with a slope of 0.
• Lines are perpendicular if the slopes are negative reciprocals; horizontal and vertical lines are always perpendicular.

Quadratic Functions

• A quadratic function is nonlinear, represented in standard form as y = ax² + bx + c, with a non-zero value of a.