Algebra 1 Flashcards: Roots and Solutions

Roots, Solutions, Zeroes

• Defined as the values of X where the equation equals zero.
• Found by factoring and setting the equation equal to zero to solve for X.

Number of Roots, Solutions, Zeroes

• Determined by the highest degree of X in the equation.
• The count matches the number of roots, solutions, or zeroes.

Degree of X

• The exponent to which X is raised, e.g., in X^2, the degree is 2.

Intercepts

• Y-Intercept: Occurs when X equals 0; also referred to as "b"; it’s where the graph crosses the y-axis.
• X-Intercept: Occurs when Y equals 0; it’s where the graph crosses the x-axis.

Slope

• Represents the steepness of a line, calculated as "Rise over Run."
• Formula for slope: (Y2 - Y1) / (X2 - X1).

Types of Equations

• Linear Equation: Highest degree is 1, represented as y = mx + b (e.g., y = 3x + 6).
• Quadratic Equation: Highest degree is 2 with terms including X^2 (e.g., y = X^2 + 3X + 7).

Equation Forms

• Slope-Intercept Form: Written as y = mx + b, where m is the slope and b is the intercept (e.g., y = 4x - 2).
• Standard Form: Represented usually as Ax + By = C (e.g., 4x - 3y + 5).

Graph Translations

• Vertical Translations: Achieved by adding/subtracting to the function, e.g., moving 5 units up translates y = X^2 to y = X^2 + 5.
• Horizontal Translations: Inside parentheses; moving right subtracts from X, left adds to X (e.g., (X - 4)^2 moves right, (X + 4)^2 moves left).

Reflections

• Over Y-Axis: Changes (X, Y) to (-X, Y).
• Over X-Axis: Changes (X, Y) to (X, -Y).

Parabola

• The graphical representation of a quadratic function, curved upwards or downwards.

Quadratic Graph Direction

• Facing Downward: When the coefficient of X^2 is negative.
• Facing Upward: When the coefficient of X^2 is positive.

Algebraic Expressions

• Binomial: An expression with two terms, e.g., (x - 2).
• Trinomial: An expression with three terms, e.g., (X^3 + 5X - 7).

FOIL Method

• A technique for multiplying two binomials: First, Outer, Inner, Last; followed by combining like terms (e.g., (x - 3)(x + 6)).

Combining Like Terms

• Process of simplifying expressions by adding terms with the same degree (e.g., 5X^2 + 7X - 4X + 10).

Positive and Negative Numbers

• Negative Numbers: Less than 0.
• Positive Numbers: Greater than 0.

Quadratic Formula

• Used when trinomials cannot be factored: X = [-b ± sqrt(b^2 - 4ac)] / (2a).

Axis of Symmetry

• Divides a quadratic graph into two mirror images.
• Calculated with x = -b/(2a); it's the x-coordinate of the vertex.

Vertex

• The highest or lowest point of a quadratic graph, representing max/min.
• Located at the x-coordinate of the Axis of Symmetry (-b/2a), with the y-coordinate found by plugging X into the equation.

Domain and Range

• Domain: Set of all possible X values (e.g., X ≥ 7).
• Range: Set of all possible Y values.