Questions and Answers
What is the additive inverse to a?
What is the multiplicative inverse to a (when a does not equal 0)?
1/a
What does raising an expression to a negative exponent equal?
1/b^n
How do you multiply expressions with powers?
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What is the result of raising expressions to powers?
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What is the formula for dividing expressions with powers?
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What do fractional exponents represent?
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What is the result of multiplying radicals?
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What is the formula for factoring the difference of two squares?
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What is the formula for factoring perfect squares?
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How is the slope of a line determined given two points?
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What is the equation of a line in slope-intercept form?
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What is the point-slope form of a line given two points?
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What is the intercept form of a line given x and y intercepts?
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What is the condition for slopes of perpendicular lines?
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How is the discriminant used to determine the number of real solutions?
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What does logarithm addition state?
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What does logarithm subtraction state?
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What is the logarithm power rule?
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What is the formula for an arithmetic sequence?
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What is the sum of arithmetic sequences?
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What is the formula for a geometric sequence?
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What is the sum of geometric sequences?
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What is the sum of infinite geometric sequences?
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What is the definition of a factorial?
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What is the combination formula of n choose r?
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What is the determinant of a 2x2 matrix represented as?
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How do you find the additive inverse of an expression?
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What is the result of raising an expression to a negative number?
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What does raising to a fractional power signify?
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How can you determine whether a graph is a function?
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How do you verify if a graph matches that of an algebraic function?
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How do you find the domain given y=f(x)?
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How do you graph a line given y=mx+b?
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What indicates whether a graph is odd, even, or neither?
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How do you describe transformations given f(x)?
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How do you determine how many roots a given graph has?
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What is the maximum and minimum number of roots for a polynomial of degree n?
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How do you determine the roots given f(x)?
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Study Notes
Algebraic Concepts
- Additive Inverse: The additive inverse of a number 'a' is -a.
- Multiplicative Inverse: The multiplicative inverse of a non-zero number 'a' is 1/a.
- Negative Exponent Rule: Raising an expression 'b' to a negative exponent 'n' is equivalent to 1/b^n.
- Exponent Addition: When multiplying expressions with the same base, x^n * x^m = x^(m+n).
- Exponent Raising: (x^m)^n simplifies to x^(m*n).
- Exponent Subtraction: For division, x^m / x^n = x^(m-n).
Advanced Exponents and Radicals
- Fractional Exponents: x^(1/2) represents the square root of x, and x^(m/n) denotes the n-th root of x^m.
- Multiplying Radicals: The product of two square roots is √a * √b = √(ab).
Factoring Techniques
- Difference of Squares: x^2 - y^2 factors into (x - y)(x + y).
- Perfect Squares: x^2 + 2xy + y^2 = (x + y)^2 and x^2 - 2xy + y^2 = (x - y)^2.
- Cubes Factoring: x^3 + y^3 = (x + y)(x^2 - xy + y^2) and x^3 - y^3 = (x - y)(x^2 + xy + y^2).
Slope and Line Equations
- Slope from Points: Given two points, the slope (m) is calculated as m = (y2 - y1) / (x2 - x1).
- Slope-Intercept Form: The equation of a line can be expressed as y = mx + b, where m is the slope and b is the y-intercept.
- Point-Slope Form: For a line through point (x1, y1) with slope m: (y - y1) = m(x - x1).
- Intercept Form: Expressed as x/a + y/b = 1, where a and b are x and y intercepts respectively.
Graphing and Properties
- Perpendicular Line Slopes: Slopes of perpendicular lines are negative reciprocals of each other.
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Discriminant: Used to determine the number of real solutions:
- If b^2 - 4ac > 0, there are 2 real solutions.
- If b^2 - 4ac = 0, there is 1 real solution.
- If b^2 - 4ac < 0, there are no real solutions but 2 imaginary solutions.
Logarithmic Operations
- Addition of Logarithms: log a + log b = log(ab).
- Subtraction of Logarithms: log a - log b = log(a/b).
- Power Rule: log a^b = b log a.
Sequences
- Arithmetic Sequence: Defined as an_n = a1 + (n - 1)d, where d is the common difference.
- Sum of Arithmetic Sequences: Calculated using Sn = n/2[2a1 + (n - 1)d] or Sn = n/2(a1 + a2).
- Geometric Sequence: An expression a_n = a1 * r^(n-1), where r is the common ratio.
- Sum of Geometric Sequences: Calculated as Sn = [a1(1 - r^n)] / (1 - r).
- Infinite Geometric Series: Sn = a1 / (1 - r), valid for |r| < 1.
Combinatorics and Linear Algebra
- Factorial: Defined as n! = n(n - 1)(n - 2)...321.
- Combination Formula: nCr = n! / (r!(n - r)!).
- Determinant of a 2x2 Matrix: For a matrix with elements a, b, c, d, the determinant is ad - bc.
Additional Concepts
- Vertical Line Test: Used to determine whether a graph represents a function.
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Domain Determination: Domain exceptions occur when:
- The denominator equals zero.
- An even root yields a negative number.
- The logarithm of zero or a negative number occurs.
- Graph Symmetry: Functions are even if symmetrical about the y-axis and odd if symmetrical about the origin.
- Transformations: Adjustments to functions include vertical shifts, horizontal shifts, and reflections across axes.
Polynomial Roots
- Root Analysis: The number of roots corresponds to how many times the graph touches or crosses the x-axis.
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Max and Min Roots:
- For a polynomial of degree n, the maximum number of real roots is n.
- If n is even, the minimum is 0; if n is odd, the minimum is 1.
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These flashcards are designed to help you master key concepts in college algebra, including inverses, exponents, and power properties. Each card provides a term along with its definition, making it easier for you to test your knowledge and prepare for the exam.