Questions and Answers
What is the definition of a relation?
What is the domain in a function?
x-coordinates, input
What is the range in a function?
y-coordinates, output
A function allows for repeated x-coordinates.
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What is (a^2)(a^6)?
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What is (a^2)^6?
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What is (a^6)/(a^2)?
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What is 2^(-3)?
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What is 1/(x^-2)?
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What is the formula for direct variation?
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What is the formula for inverse variation?
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What is the formula for joint variation?
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What do horizontal asymptotes depend on in rational expressions?
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How do you find the x-intercepts for rational expressions?
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What does continuous compounding depend on?
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What is the square root of -1?
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What is i^2?
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What does a rational exponent mean?
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How do you find the inverse of a function?
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How do you solve an absolute value equation?
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What ways can you solve a system of equations?
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What is the Compound Interest Formula?
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What is the Standard Form for a Quadratic?
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What is the Vertex Form for a Quadratic?
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What is the Intercept Form for a Quadratic?
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What is the Standard Form for a Line?
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What is the Slope-Intercept Form for a Line?
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What is the Slope Formula?
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What is the Point-Slope Formula?
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What is the Quadratic Formula?
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What is Completing the Square?
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What methods can be used to solve a quadratic?
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What is the Standard Form Equation for a Circle with center (h, k)?
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What is the Standard Form Equation for a Circle with center at the origin?
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What is the domain for an exponential function?
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What is the range for a logarithmic function?
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What is a matrix?
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What must be true to add two matrices?
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What must be true to multiply two matrices?
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Study Notes
Algebra II Key Concepts
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Relation: Defined as a set of ordered pairs, representing connections between variables.
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Domain: Refers to the x-coordinates (input values) of a function.
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Range: Represents the y-coordinates (output values) of a function.
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Function: Must have distinct x-values; can be validated using the vertical line test on its graph.
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Exponent Rules:
- ( (a^2)(a^6) = a^8 )
- ( (a^2)^6 = a^{12} )
- ( \frac{a^6}{a^2} = a^4 )
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Negative Exponents:
- ( 2^{-3} = \frac{1}{2^3} = \frac{1}{8} )
- ( \frac{1}{x^{-2}} = x^2 )
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Variation Types:
- Direct Variation: Expressed as ( y = kx )
- Inverse Variation: Expressed as ( y = \frac{k}{x} )
- Joint Variation: Expressed as ( y = kxz ), indicating dependence on multiple variables.
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Asymptotes in Rational Expressions:
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Horizontal Asymptotes:
- If degree of the denominator > degree of numerator, then ( y = 0 ).
- If degrees are the same, use ( \frac{a}{c} ) where ( a ) and ( c ) are leading coefficients.
- If the numerator's degree > denominator's, there is no horizontal asymptote.
- Vertical Asymptotes: Set the denominator to zero to find potential asymptotes, ensuring common factors are divided out.
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Horizontal Asymptotes:
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Finding x-intercepts: Set the numerator of a rational expression to zero and factor, remembering to reduce by any common factors from the denominator.
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Sum/Difference of Cubes: Factoring structure includes first term, second term, first term squared, product of the first and second term, last term squared; signs follow: first retains its sign, second is opposite, and the last is always plus.
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Continuous Compounding Formula: ( A = Pe^{rt} ), where ( P ) is the principal, ( r ) is the rate, and ( t ) is time.
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Imaginary Unit: The square root of -1 is represented as ( i ); thus ( i^2 = -1 ).
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Rational Exponents: Indicates a power over a root format, combining exponent rules with roots.
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Finding Inverses: To find the inverse of a function:
- Rewrite in ( y = ) format.
- Switch x and y.
- Solve for y.
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Solving Absolute Value Equations: Set the expression inside the absolute value equal to both the positive and negative constants.
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System of Equations Resolution Methods: Includes graphing, substitution, elimination (linear combination), and matrix equations.
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Inequalities for Absolute Values: Use GOLA; "Greater than" implies OR, "Less than" implies AND.
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Compound Interest Formula: ( A = P(1 + \frac{r}{n})^{nt} ), detailing the variables of principal, rate, and time compounded ( n ) times.
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Quadratic Forms:
- Standard Form: ( f(x) = ax^2 + bx + c )
- Vertex Form: ( y = a(x - h)^2 + k )
- Intercept Form: ( y = a(x - p)(x - q) )
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Linear Equations:
- Standard Form: ( Ax + By = C )
- Slope-Intercept Form: ( y = mx + b )
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Slope Calculation: Calculated as ( m = \frac{y_2 - y_1}{x_2 - x_1} ), indicating the ratio of change in y to change in x.
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Point-Slope Formula: ( y - y_1 = m(x - x_1) ), used to establish a line's equation given a point and slope.
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Quadratic Formula: ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ) utilized for finding roots.
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Completing the Square: Requires a leading coefficient of 1. Take half of ( b ), square it, add to both sides, and solve through square roots.
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Quadratic Solution Methods: Include graphing, factoring (if possible), completing the square, and the quadratic formula.
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Circle Equations:
- Center at ( (h, k) ): ( (x - h)^2 + (y - k)^2 = r^2 )
- Center at origin: ( x^2 + y^2 = r^2 )
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Exponential Function Domain: Includes all real numbers.
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Logarithmic Function Range: Encompasses all real numbers.
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Matrix Definition: A rectangular array of numbers organized in rows and columns.
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Matrix Addition Condition: Only possible if dimensions (rows and columns) match.
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Matrix Multiplication Condition: Requires that the number of columns in the first matrix equals the number of rows in the second matrix.
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Test your knowledge on key concepts in Algebra II, including relations, domain, range, and function characteristics. Additionally, challenge yourself with exponent rules and types of variation. This quiz covers essential topics necessary for mastering algebra.