CLEP College Algebra Flashcards
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Questions and Answers

What is the additive inverse to a?

  • expression^1/n
  • 1/a
  • 1/(expression)^n
  • -a (correct)

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?

<p>x^m+n</p> Signup and view all the answers

What is the result of raising expressions to powers?

<p>(x^m)^n = x^mn</p> Signup and view all the answers

What is the formula for dividing expressions with powers?

<p>x^m-n</p> Signup and view all the answers

What do fractional exponents represent?

<p>x^1/2 = square root of x, x^m/n = n-th root of x^m</p> Signup and view all the answers

What is the result of multiplying radicals?

<p>(sqrt a)(sqrt b) = sqrt(ab)</p> Signup and view all the answers

What is the formula for factoring the difference of two squares?

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

What is the formula for factoring perfect squares?

<p>x^2 + 2xy + y^2 = (x + y)^2, x^2 - 2xy + y^2 = (x - y)^2</p> Signup and view all the answers

How is the slope of a line determined given two points?

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

What is the equation of a line in slope-intercept form?

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

What is the point-slope form of a line given two points?

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

What is the intercept form of a line given x and y intercepts?

<p>x/a + y/b = 1</p> Signup and view all the answers

What is the condition for slopes of perpendicular lines?

<p>inverses of each other</p> Signup and view all the answers

How is the discriminant used to determine the number of real solutions?

<p>b^2 - 4ac</p> Signup and view all the answers

What does logarithm addition state?

<p>log a + log b = log(ab)</p> Signup and view all the answers

What does logarithm subtraction state?

<p>log a - log b = log(a/b)</p> Signup and view all the answers

What is the logarithm power rule?

<p>log a^b = b log a</p> Signup and view all the answers

What is the formula for an arithmetic sequence?

<p>an = a1 + (n - 1)d</p> Signup and view all the answers

What is the sum of arithmetic sequences?

<p>Sn = n/2 [2a1 + (n - 1)d] or Sn = n/2 (a1 + a2)</p> Signup and view all the answers

What is the formula for a geometric sequence?

<p>an = a1r^(n-1)</p> Signup and view all the answers

What is the sum of geometric sequences?

<p>Sn = [a1(1 - r^n)] / (1 - r)</p> Signup and view all the answers

What is the sum of infinite geometric sequences?

<p>Sn = a1 / (1 - r)</p> Signup and view all the answers

What is the definition of a factorial?

<p>n! = n(n - 1)(n - 2)...3(2)(1)</p> Signup and view all the answers

What is the combination formula of n choose r?

<p>nCr = n! / (r!(n - r)!)</p> Signup and view all the answers

What is the determinant of a 2x2 matrix represented as?

<p>ad - bc</p> Signup and view all the answers

How do you find the additive inverse of an expression?

<p>It is the negative of the expression.</p> Signup and view all the answers

What is the result of raising an expression to a negative number?

<p>1 / (expression)^n</p> Signup and view all the answers

What does raising to a fractional power signify?

<p>expression^(1/n) = n-th root(expression)</p> Signup and view all the answers

How can you determine whether a graph is a function?

<p>Using the vertical line test</p> Signup and view all the answers

How do you verify if a graph matches that of an algebraic function?

<p>Plug x-values into the function to find the values of y.</p> Signup and view all the answers

How do you find the domain given y=f(x)?

<p>The domain will be infinite with three exceptions.</p> Signup and view all the answers

How do you graph a line given y=mx+b?

<p>Choose two values for x and find the values of y.</p> Signup and view all the answers

What indicates whether a graph is odd, even, or neither?

<p>Even if symmetrical with respect to the y-axis, odd if symmetrical with respect to the origin.</p> Signup and view all the answers

How do you describe transformations given f(x)?

<p>f(x) + a is a units up, f(x) - a is a units down, f(x - a) is a units to the right, f(x + a) is a units to the left, -f(x) reflects across the x-axis.</p> Signup and view all the answers

How do you determine how many roots a given graph has?

<p>Count how many times the graph touches or crosses the x-axis.</p> Signup and view all the answers

What is the maximum and minimum number of roots for a polynomial of degree n?

<p>Max number of real roots is n; min is 0 if n is even, and 1 if n is odd.</p> Signup and view all the answers

How do you determine the roots given f(x)?

<p>Set f(x) = 0 and factor f(x).</p> Signup and view all the answers

Flashcards

Additive Inverse

The opposite of a number, which when added to the original number, results in zero.

Multiplicative Inverse

The reciprocal of a non-zero number, which when multiplied by the original number, equals 1.

Negative Exponent

For b⁻ⁿ, it's 1/bⁿ.

Exponent Addition

When multiplying expressions with the same base, add exponents: xⁿ * xᵐ = x^(n+m).

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Exponent Raising

(xᵐ)ⁿ = x^(m*n).

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Exponent Subtraction

When dividing expressions with the same base, subtract exponents: xᵐ / xⁿ = x^(m-n).

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Fractional Exponents

x^(m/n) means the nth root of x to the m power.

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Multiplying Radicals

√a * √b = √(ab).

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Difference of Squares

x² - y² = (x-y)(x+y).

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Perfect Squares

x² + 2xy + y² = (x+y)² and x² - 2xy + y² = (x-y)²

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Cubes Factoring

x³ + y³ = (x+y)(x² -xy + y²) and x³ - y³ = (x-y)(x² +xy + y²).

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Slope from Points

m = (y₂ - y₁)/(x₂ - x₁).

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Slope-Intercept Form

y = mx + b, where m is slope, b is y-intercept.

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Point-Slope Form

(y - y₁) = m(x - x₁).

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Intercept Form

x/a + y/b = 1.

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Perpendicular Lines

Slopes of perpendicular lines are negative reciprocals.

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Discriminant

b² - 4ac, determines the number of real roots in a quadratic equation.

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Logarithm Addition

log a + log b = log(ab).

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Logarithm Subtraction

log a - log b = log(a/b).

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Power Rule (Logs)

log aᵇ = b log a.

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Arithmetic Sequence

aₙ = a₁ + (n-1)d

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Sum of Arithmetic Sequence

Sₙ = n/2[2a₁ + (n-1)d]

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Geometric Sequence

aₙ = a₁ * r^(n-1)

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Sum of Geometric Sequence

Sₙ = [a₁(1 - rⁿ)] / (1 - r)

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Infinite Geometric Series

S = a₁ / (1 - r), |r| < 1

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Factorial

n! = n(n-1)(n-2)...321

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Combination

nCr = n! / (r! (n-r)!)

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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.
  • 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.
  • 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.
  • 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.

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