Algebra Review

Review of Algebra including exponents, radicals, logarithms, factoring, algebraic fractions, equations, inequalities, graphing and functions

What is the base in the exponential expression b^n?

b

What is the exponent or power in the exponential expression b^n?

n

What does bn equal if n is a positive integer?

b•b•b•b••••b (n times)

What does b^0 equal?

1 (where b≠0)

What does b-n equal?

1/bn (where b≠0)

What does n√b equal?

b1/n (where n≠0, and if n is even, then b≥0

What does n√bm equal?

(n√b)m = (b1/n)m = bm/n n≠0, and if n is even, then b≥0

When multiplying like bases, what do you do with the exponents?

Add them: bn • bm = bn+m

When dividing like bases, what do you do with the exponents?

Subtract them: bn / bm = bn-m

What do you do with exponents when you have an exponent of an exponent?

Multiply them: (bn)m = bn•m

How do you remove parentheses in the expression (ab)n?

(ab)n = an • bn

How do you remove parentheses in the expression (a/b)n?

(a/b)n = an / bn

What is the convention for representing the base 10 logarithm log10A = n?

logA = n

What is the base of the natural logarithm?

e (e ~ 2.718)

What is the convention for abbreviating loge A= n?

lnA = n

What is logb 1 equal to?

0

What is logb b equal to?

1

What is the first law of logarithms?

logb x + logb y = logb (x • y)

What is the second law of logarithms?

logb x – logb y = logb (x / y)

What is the third law of logarithms?

n•logb x = logb xn

What is the distributive law?

ax + ay = a(x + y)

What is a simple trinomial?

x2 +(a+b)x + a•b= (x + a)(x + b)

What is the difference of squares?

x2 – a2 = (x–a)(x+a)

What is the sum of cubes?

x3 +a3 = (x+a)(x2 – ax+a2 )

What is the difference of cubes?

x3 – a3 = (x–a)(x2 + ax+a2 )

Write the quadratic formula for the equation ax2 + bx + c = 0.

x = (– b ± √(b2 – 4ac)) / (2a)

How do you add fractions?

Find a common denominator: a/b + c/d = (ad+bc) / bd

How do you subtract fractions?

Find a common denominator: a/b – c/d = (ad– bc) / bd

How do you multiply fractions?

(a/b) • (c/d) = ac / bd

How do you divide fractions?

Invert and multiply: (a/b) / (c/d) = (a/b) • (d/c) = ad / bc

What is the procedure to solve first degree equations?

First degree equations are solved using addition, subtraction, multiplication and division.

What is the procedure to solve second degree or quadratic equations?

Second degree equations or quadratic equations are solved by factoring or the quadratic formula.

How are Equations involving absolute value solved?

Equations involving absolute value are equivalent to two equations without the absolute value sign.

When solving inequalities, what happens when multiplying or dividing both sides of an inequality by negative numbers?

Requires the inequality sign to be reversed.

What is the simplest way to begin any graph?

Plot the x- and y-intercepts (if any).

What are the key graphing steps involved in graphing Lines Ax +By + C = 0?

Find the y-intercept (let x = 0 and solve for y) and the x-intercept (let y = 0 and solve for x) and draw a line through them.

What are the key graphing steps involved in graphing Absolute Values y - k = |x – h|?

Find your intercepts then graph its vertex. This is simply the point (h,k), but you must be sure your equation is in the form given above.

What are the key graphing steps involved in graphing Parabolas y = ax2 + bx + c?

After finding the x- and y-intercepts, you must also find the vertex. The x-coordinate of the vertex, Vx , may be found three ways: 1. Vx = (x1 + x2)/2. 2. Vx = – b / 2a 3. Complete the square

What are the key graphing steps involved in graphing Circles (x – h)2 + (y – k)2 = r 2?

In addition to the x- and y-intercepts, you must plot the center (h,k) and demonstrate the length of the radius,r. If your equation is not in this form you must complete the square to put it in the correct form.

How do you translate the a function when given y = f(x) + C?

Moved up C units from y = f(x).

How do you translate the a function when given y = f(x) – C?

Moved down C units from y = f(x).

How do you translate the a function when given y = f(x – C)?

Moved to the right C units from y = f(x).

How do you translate the a function when given y = f(x + C)?

Moved to the left C units from y = f(x).

What constitutes a function

A relationship between two variables where each value of the independent variable only corresponds exactly to one value of the dependent variable.

Briefly describe the Domain of a function.

All of the values of the independent variable for which its function is defined.

Name a domain rule involving the variable 'k'

You cannot divide by zero, k/0

Name a domain rule involving square roots.

You cannot take the even root of a negative number

Name a domain rule involving logarithms.

You cannot take the logarithm of a negative or of zero.

Briefly describe the range of a function.

All of the values of the dependent variable.

What is the product of the composition of a function (f ° g)(x)?

Directing you to first find g(x). The result of g(x) is then put in to the function f(x).

What are two functions, f(x) and g(x), called if the components of every ordered pair of f(x) are in interchanged positions in the function g(x) and vice versa?

Inverses