Algebra II Key Concepts
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Questions and Answers

What is the definition of a relation?

  • an equation
  • a variable
  • a function
  • set of ordered pairs (correct)

    • 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.

    <p>False</p> Signup and view all the answers

    What is (a^2)(a^6)?

    <p>a^8</p> Signup and view all the answers

    What is (a^2)^6?

    <p>a^12</p> Signup and view all the answers

    What is (a^6)/(a^2)?

    <p>a^4</p> Signup and view all the answers

    What is 2^(-3)?

    <p>1/(2^3) = 1/8</p> Signup and view all the answers

    What is 1/(x^-2)?

    <p>x^2</p> Signup and view all the answers

    What is the formula for direct variation?

    <p>y = kx</p> Signup and view all the answers

    What is the formula for inverse variation?

    <p>y = k/x</p> Signup and view all the answers

    What is the formula for joint variation?

    <p>y = kxz</p> Signup and view all the answers

    What do horizontal asymptotes depend on in rational expressions?

    <p>if degree of Bottom is bigger, y=0; if degrees are same for numerator and denominator, use coefficients a/c; if Top has a larger degree, NO horizontal asymptote</p> Signup and view all the answers

    How do you find the x-intercepts for rational expressions?

    <p>set numerator equal to zero and factor</p> Signup and view all the answers

    What does continuous compounding depend on?

    <p>A = Pe^(rt)</p> Signup and view all the answers

    What is the square root of -1?

    <p>i</p> Signup and view all the answers

    What is i^2?

    <p>-1</p> Signup and view all the answers

    What does a rational exponent mean?

    <p>power/root = power over root</p> Signup and view all the answers

    How do you find the inverse of a function?

    <ol> <li>'y = ' form, 2. switch x and y, 3. solve for y</li> </ol> Signup and view all the answers

    How do you solve an absolute value equation?

    <p>Set the expression from inside the absolute value equal to the positive constant and to the negative constant.</p> Signup and view all the answers

    What ways can you solve a system of equations?

    <ol> <li>graph, 2. substitution, 3. linear combination or elimination, 4. use a matrix equation if the system is linear</li> </ol> Signup and view all the answers

    What is the Compound Interest Formula?

    <p>A = P(1 + r/n)^(nt)</p> Signup and view all the answers

    What is the Standard Form for a Quadratic?

    <p>f(x) = ax^2 + bx + c</p> Signup and view all the answers

    What is the Vertex Form for a Quadratic?

    <p>y = a(x - h)^2 + k</p> Signup and view all the answers

    What is the Intercept Form for a Quadratic?

    <p>y = a(x - p)(x - q)</p> Signup and view all the answers

    What is the Standard Form for a Line?

    <p>Ax + By = C</p> Signup and view all the answers

    What is the Slope-Intercept Form for a Line?

    <p>y = mx + b</p> Signup and view all the answers

    What is the Slope Formula?

    <p>m = (y2 - y1)/(x2 - x1)</p> Signup and view all the answers

    What is the Point-Slope Formula?

    <p>y - y1 = m(x - x1)</p> Signup and view all the answers

    What is the Quadratic Formula?

    <p>x = (-b ± sqrt(b^2 - 4ac)) / 2a</p> Signup and view all the answers

    What is Completing the Square?

    <p>Have a coefficient of 1 for the squared term, take 1/2 of b and square it, add to both sides, solve through square roots.</p> Signup and view all the answers

    What methods can be used to solve a quadratic?

    <ol> <li>Graphing, 2. Factoring, 3. Completing the Square, 4. Quadratic Formula</li> </ol> Signup and view all the answers

    What is the Standard Form Equation for a Circle with center (h, k)?

    <p>(x - h)^2 + (y - k)^2 = r^2</p> Signup and view all the answers

    What is the Standard Form Equation for a Circle with center at the origin?

    <p>x^2 + y^2 = r^2</p> Signup and view all the answers

    What is the domain for an exponential function?

    <p>All real numbers</p> Signup and view all the answers

    What is the range for a logarithmic function?

    <p>All real numbers</p> Signup and view all the answers

    What is a matrix?

    <p>Rectangular array of numbers in rows and columns</p> Signup and view all the answers

    What must be true to add two matrices?

    <p>The dimensions must be the same (Rows and Columns)</p> Signup and view all the answers

    What must be true to multiply two matrices?

    <p>The columns of the first must match the rows of the second matrix</p> Signup and view all the answers

    Study Notes

    Algebra II Key Concepts

    • Relation: Defined as a set of ordered pairs, representing connections between variables.

    • Domain: Refers to the x-coordinates (input values) of a function.

    • Range: Represents the y-coordinates (output values) of a function.

    • Function: Must have distinct x-values; can be validated using the vertical line test on its graph.

    • Exponent Rules:

      • ( (a^2)(a^6) = a^8 )
      • ( (a^2)^6 = a^{12} )
      • ( \frac{a^6}{a^2} = a^4 )

    • Negative Exponents:

      • ( 2^{-3} = \frac{1}{2^3} = \frac{1}{8} )
      • ( \frac{1}{x^{-2}} = x^2 )

    • 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.

    • Asymptotes in Rational Expressions:

      • 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.

    • Finding x-intercepts: Set the numerator of a rational expression to zero and factor, remembering to reduce by any common factors from the denominator.

    • 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.

    • Continuous Compounding Formula: ( A = Pe^{rt} ), where ( P ) is the principal, ( r ) is the rate, and ( t ) is time.

    • Imaginary Unit: The square root of -1 is represented as ( i ); thus ( i^2 = -1 ).

    • Rational Exponents: Indicates a power over a root format, combining exponent rules with roots.

    • Finding Inverses: To find the inverse of a function:

      • Rewrite in ( y = ) format.
      • Switch x and y.
      • Solve for y.

    • Solving Absolute Value Equations: Set the expression inside the absolute value equal to both the positive and negative constants.

    • System of Equations Resolution Methods: Includes graphing, substitution, elimination (linear combination), and matrix equations.

    • Inequalities for Absolute Values: Use GOLA; "Greater than" implies OR, "Less than" implies AND.

    • Compound Interest Formula: ( A = P(1 + \frac{r}{n})^{nt} ), detailing the variables of principal, rate, and time compounded ( n ) times.

    • 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) )

    • Linear Equations:

      • Standard Form: ( Ax + By = C )
      • Slope-Intercept Form: ( y = mx + b )

    • 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.

    • Point-Slope Formula: ( y - y_1 = m(x - x_1) ), used to establish a line's equation given a point and slope.

    • Quadratic Formula: ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ) utilized for finding roots.

    • Completing the Square: Requires a leading coefficient of 1. Take half of ( b ), square it, add to both sides, and solve through square roots.

    • Quadratic Solution Methods: Include graphing, factoring (if possible), completing the square, and the quadratic formula.

    • Circle Equations:

      • Center at ( (h, k) ): ( (x - h)^2 + (y - k)^2 = r^2 )
      • Center at origin: ( x^2 + y^2 = r^2 )

    • Exponential Function Domain: Includes all real numbers.

    • Logarithmic Function Range: Encompasses all real numbers.

    • Matrix Definition: A rectangular array of numbers organized in rows and columns.

    • Matrix Addition Condition: Only possible if dimensions (rows and columns) match.

    • 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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    Description

    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.

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