Solving Exponential Equations, where x is in the exponent, BUT the bases DO NOT MATCH.
Step 1: Isolate the exponential expression.
Get your exponential expression on one side everything outside of the exponential expression on the other side of your equat
Slope
The slope of a line measures the steepness of the line. Most of you are probably familiar with associating slope with "rise over run". ny units you move up or down from point to point. On the graph that would be a change in the y values. Run means h
Rational Expression A rational expression is one that can be written in the form
where P and Q are polynomials and Q does not equal 0.
An example of a rational expression is:
Domain of a Rational Expression
With rational functions, we need to watch out fo
Rational Inequalities A rational inequality is one that can be written in one of the following standard forms:
or
or
or
Q does not equal 0.
In other words, a rational inequality is in standard form when the inequality is set to 0.
Solving Rational Inequal
Quadratic Inequalities A quadratic inequality is one that can be written in one of the following standard forms:
or or or
In other words, a quadratic inequality is in standard form when the inequality is set to 0. Just like in a quadratic equation, the de
Quadratic in Form An equation is quadratic in form when it can be written in this standard form
where the same expression is inside both ( )'s.
In other words, if you have a times the square of the expression following b plus b times that same expression
Quadratic Function
A quadratic function is a function that can be written in the form
where a, b, and c are constants and
Note that in a quadratic function there is a power of two on your independent variable and that is the highest power.
Standard Form o
Quadratic Equation
Standard form:
, Where a does not equal 0. Note that in Tutorial 14: Linear Equations in One Variable, we learned that a linear equation can be written in the form ax + b = 0 and that the exponent on the variable was 1. Note that the di
Polynomial Function
A polynomial function is a function that can be written in the form , where are real numbers and n is a nonnegative integer.
Basically it is a function whose rule is given by a polynomial in one variable. If you need a review on functi
Polynomial Equation
A polynomial equation is one polynomial set equal to another polynomial. If you need a review on polynomials, feel free to go to Tutorial 6: Polynomials. The following is an example of a polynomial equation:
Standard Form of a Polynomi
Parallel Lines and Their Slopes
In other words, the slopes of parallel lines are equal. Note that two lines are parallel if their slopes are equal and they have different yintercepts.
Perpendicular Lines and Their Slopes
In other words, perpendicular slop
Solving Radical Equations
Step 1: Isolate one of the radicals.
get one radical on one side and everything else on the other using inverse operations. In some problems there is only one radical. However, there are some problems that have more than one radi
Solving Rational Equations
Step 1: Simplify by removing the fractions.
ultiplying both sides by the LCD. If you need a review on finding the LCD of a rational expression, go to Tutorial 10: Adding and Subtracting Rational Expressions. Note that even thoug
Properties of Logarithms
As mentioned above - and I cant emphasize this enough - logs are another way to write exponents. If you understand that concept it really does make things more pleasant when you are working with logs.
Property 1 Product Rule
m > 0
Definition of Log Function
The logarithmic function with base b, where b > 0 and b 1, is denoted by and is defined by
if and only if
IN OTHER WORDS - AND I CAN NOT STRESS THIS ENOUGH- A LOG IS ANOTHER WAY TO WRITE AN EXPONENT. This definition can work in
Solving a Logarithmic Equation of the Form
Step 1: Write as one log isolated on one side.
Get your log on one side everything outside of the log on the other side of your equation using inverse operations. Also use properties of logs to write it so that t
Definition of Exponential Function
The function f defined by
where b > 0, b 1, and the exponent x is any real number, is called an exponential function.
Again, note that the variable x is in the exponent as opposed to the base when we are dealing with an
Exponential Growth
, (k > 0)
A represents the amount at a given time t. Ao represents the initial amount of the growing entity. Note that this is the amount when t = 0. k is a constant that represents the growth rate. It is POSITIVE when talking in terms
The Upper and Lower Bound Theorem
Upper Bound
If you divide a polynomial function f(x) by (x - c), where c > 0, using synthetic division and this yields all positive numbers, then c is an upper bound to the real roots of the equation f(x) = 0. Note that t
Rational Zero (or Root) Theorem If , where is a rational zero,
are integer coefficients and the reduced fraction
then p is a factor of the constant term coefficient
and q is a factor of the leading .
We can use this theorem to help us find all of the POSS
Translating an English Phrase Into an Algebraic Expression
Sometimes, you find yourself having to write out your own algebraic expression based on the wording of a problem. In that situation, you want to
1. read the problem carefully, 2. pick out key word
The Standard Form of the Equation of a Circle
(h, k) is the center r is the radius (x, y) is any point on the circle
All points (x, y) on the circle are a fixed distance (radius) away from the center (h, k). The h value of your center is the first value o
Synthetic Division
Synthetic division is another way to divide a polynomial by the binomial x - c , where c is a constant.
Step 1: Set up the synthetic division.
An easy way to do this is to first set it up as if you are doing long division and then set u
The following show us how to perform the different operations on functions.
Use the functions to illustrate the operations:
and
Sum of f + g (f + g)(x) = f(x) + g(x)
This is a very straight forward process. When you want the sum of your functions you simp
Multiplying Rational Expressions
Q and S do not equal 0. Step 1: Factor both the numerator and the denominator.
If you need a review on factoring, feel free to go back to Tutorial 7: Factoring Polynomials.
Step 2: Write as one fraction.
Write it as a prod
Divide Polynomial Polynomial
Using Long Division Step 1: Set up the long division.
at you are dividing by) goes on the outside of the box. The dividend (what you are dividing into) goes on the inside of the box. When you write out the dividend, make sure
Adding or Subtracting Rational Expressions with Common Denominators
Step 1: Combine the numerators together. Step 2: Put the sum or difference found in step 1 over the common denominator. Step 3: Reduce to lowest terms as shown in Tutorial 8: Simplifying
Absolute Value
A lot of people know that when you take the absolute value of a number the answer is positive, but do you know why? Let's find out: The absolute value measures the DISTANCE a number is away from the origin (zero) on the number line. No matt
Scientific Notation A positive number is written in scientific notation if it is written in the form:
where 1 < a < 10 and r is an integer power of 10.
Writing a Number in Scientific Notation
Step 1: Move the decimal point so that you have a number that i
Rational Exponents and Roots
If x is positive, p and q are integers and q is positive,
In other words, when you have a rational exponent, the denominator of that exponent is your index or root number and the numerator of the exponent is the exponential pa