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?

x^m+n Signup and view all the answers

What is the result of raising expressions to powers?

(x^m)^n = x^mn Signup and view all the answers

What is the formula for dividing expressions with powers?

x^m-n Signup and view all the answers

What do fractional exponents represent?

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

What is the result of multiplying radicals?

(sqrt a)(sqrt b) = sqrt(ab) Signup and view all the answers

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

x^2 - y^2 = (x - y)(x + y) Signup and view all the answers

What is the formula for factoring perfect squares?

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

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

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

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

y = mx + b Signup and view all the answers

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

(y - y1) = m(x - x1) Signup and view all the answers

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

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

What is the condition for slopes of perpendicular lines?

inverses of each other Signup and view all the answers

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

b^2 - 4ac Signup and view all the answers

What does logarithm addition state?

log a + log b = log(ab) Signup and view all the answers

What does logarithm subtraction state?

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

What is the logarithm power rule?

log a^b = b log a Signup and view all the answers

What is the formula for an arithmetic sequence?

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

What is the sum of arithmetic sequences?

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

What is the formula for a geometric sequence?

an = a1r^(n-1) Signup and view all the answers

What is the sum of geometric sequences?

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

What is the sum of infinite geometric sequences?

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

What is the definition of a factorial?

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

What is the combination formula of n choose r?

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

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

ad - bc Signup and view all the answers

How do you find the additive inverse of an expression?

It is the negative of the expression. Signup and view all the answers

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

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

What does raising to a fractional power signify?

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

How can you determine whether a graph is a function?

Using the vertical line test Signup and view all the answers

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

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

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

The domain will be infinite with three exceptions. Signup and view all the answers

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

Choose two values for x and find the values of y. Signup and view all the answers

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

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

How do you describe transformations given f(x)?

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. Signup and view all the answers

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

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

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

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

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

Set f(x) = 0 and factor f(x). Signup and view all the answers

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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Description

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.