Algebra Placement Test Flashcards

Properties of Exponents

• Product of Powers Property: When multiplying two powers with the same base, add their exponents: ( a^m \cdot a^n = a^{m+n} ).
• Quotient of Powers Property: When dividing two powers with the same base, subtract the exponents: ( \frac{a^m}{a^n} = a^{m-n} ).
• Power of a Power Property: To raise a power to another power, multiply the exponents: ( (a^m)^n = a^{mn} ).
• Power of a Product Property: Distribute the exponent to each factor in the product: ( (ab)^n = a^n \cdot b^n ).
• Power of a Quotient Property: Distribute the exponent to both the numerator and the denominator: ( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} ).
• Zero Exponent: Any non-zero base raised to the power of zero equals one: ( a^0 = 1 ) (where ( a \neq 0 )).
• Negative Exponent: A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent: ( a^{-n} = \frac{1}{a^n} ).
• Rational Exponent: An exponent that is a fraction indicates roots: ( a^{\frac{m}{n}} = \sqrt[n]{a^m} ).

Exponential Growth and Decay

• Exponential Growth Model: Represented by the formula ( y = \text{initial amount} \cdot (1 + \text{rate})^{\text{time}} ).
• Exponential Decay Model: Represented by the formula ( y = \text{initial amount} \cdot (1 - \text{rate})^{\text{time}} ).

Algebraic Manipulation

• Distributive Property of Addition: Shows how to distribute a multiplier across a sum: ( a(b + c) = ab + ac ).
• Associative Property of Addition: The way numbers are grouped does not affect the sum: ( (a+b) + c = a + (b+c) ).
• Associative Property of Multiplication: The grouping of numbers does not affect the product: ( (a \cdot b) \cdot c = a \cdot (b \cdot c) ).
• Addition Property of Zero: Adding zero to any number leaves it unchanged: ( a + 0 = a ).
• Distributive Property of Subtraction: Distributes across a difference: ( a(b - c) = ab - ac ).

Graphing and Data Interpretation

• Line Graph: Displays data changes over time with points connected by lines.
• Bar Graph: Uses rectangular bars to represent discrete categories or values.
• Pictograph: Uses pictures or symbols to visually represent data quantities.
• Pie Chart: Circular chart divided into slices to illustrate numerical proportions.
• Stem and Leaf Plot: Displays quantitative data in a graphical format, preserving original data values.

Rates of Change

• Rate of Change Formula: Defined as the change in y over the change in x: ( \text{Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1} ).

Comparing Models

• Linear Graph: Represents a constant rate of change in a straight line.
• Exponential Graph: Shows rapid increases or decreases, generally curved.
• Quadratic Graph: Takes the shape of a parabola, representing squared relationships.

Writing Equations of a Line

• Equation Format: ( y = mx + b ), where ( m ) is the slope and ( b ) is the y-intercept.

Graphing Lines

• Slope-Intercept Form: Focused on the slope and y-intercept for graphing linear equations.
• Point-Slope Form: Expresses the equation of a line in relation to a specific point on the graph.

Solving Quadratic Equations

• Factoring Approach: Represents a quadratic in the form ( ax^2 + bx + c ); ( b ) represents the sum of roots, and ( c ) represents the product of roots.
• Quadratic Formula: Used to find solutions for ( ax^2 + bx + c = 0 ).
• Perfect Square Trinomial: Format ( a^2 + 2ab + b^2 = (a+b)^2 ) or ( a^2 - 2ab + b^2 = (a-b)^2 ).
• Difference of Two Squares: Expresses as ( a^2 - b^2 = (a + b)(a - b) ).

Graphing Quadratic Functions

• Focuses on graphing parabolas and identifying key features such as vertex and axis of symmetry.

Test your knowledge of exponents with these flashcards covering key properties such as the product of powers and power of a product. Each card presents a fundamental concept necessary for mastering algebra. Ideal for students preparing for an algebra placement test.