Please keep in mind that any notes, videos, and calculators provided on this page are no substitute for attending class. These resources are here to supplement the information from your instructor and textbook. First and foremost please be sure to follow any directions and use any resources your teacher provides.
| SYMBOL | MEANING |
|---|---|
| x | Most commonly used variable, represents an unknown value |
| ≠ | Not equal to, does not equal |
| ≈ | Approximately equal to |
| > | Greater than |
| < | Less than |
| ≥ | Greater than or equal to |
| ≤ | Less than or equal to |
| () | Parentheses |
| [] | Brackets |
| {} | Braces |
| ± | Plus or minus |
| n√a | nth root of a |
| ∅ | Empty set |
| |a| | Absolute value of a |
| | | Such that (used in set builder notation) |
| ∪ | Union ("or") |
Click to download all chapter 1 notes (ZIP file with 8 PDFs—one per section).
You'll also find "quick notes ", like common formulas and definitions, for this chapter below. But for more thorough notes and examples please download the PDF.
Linear equations are vital in many real-world scenarios. For instance, you're in this class right now, in the real world, and you have to learn about linear equations.
Why have an equation if not to solve it?
Five steps for problem solving:
I = PRT, where I = interest, P = principal, R = interest rate, and T = time. d = rt, where d = distance, r = rate (speed), and t = time. C(x) = mx + b, where C(x) = total cost, x = number of units, m = cost per unit, and b = fixed cost.
Once upon a time mathematicians realized you couldn't take the square root of a negative number. This was a total bummer because that left a lot of problems unsolved. But then Girolamo Cardano was like, "Bro, look, we can do this thing and it will work." And René Descartes said, "What are you going to do, make something up? Oooh look at my 'iMaGiNaRy' numbers." But Cardano was just like "Yep." And Descartes was like, "... Okay fine, I'll go with it but I'm not happy about it," and now we have the imaginary unit.*
*Some liberties were taken in the writing of this historical account. But if you're interested in the full story, visit the SSC library to learn more!
Click image to enlarge.
"There is nothing in the world quite so satisfying as a quadratic equation." — said somebody, probably, at some point.
You weren't expecting geometry, were you?
Where "a" and "b" are the legs and "c" is the hypotenuse of a right triangle.
Is that boss level music?
Things don't always balance out. But it's okay, we have math for that.
Important!
When multiplying or dividing both sides of an inequality by a negative you must reverse the inequality symbol.
This is the |absolute| best. Ba dum tss.
| Absolute value equation or inequality where k ≠ 0 | Equivalent form | Solution set |
| |x| = k | x = k or x = -k | {-k, k} |
| |x| < k | -k < x < k | (-k, k) |
| |x| > k | x < -k or x > k | (-∞, -k) ∪ (k, ∞) |
The zeros of a polynomial are the x-values that, when plugged in, make the polynomial equal to zero. Several theorems exist to aid in finding all the zeros of a polynomial:
Statement:
If k is a zero of a polynomial f(x), then (x - k) is a factor of f(x), and vice versa.
Use:
If you find a zero, you can use it to factor the polynomial and find other zeros.
Statement:
If a polynomial f(x) with integer coefficients has a rational zero p/q (where p and q are integers with no common factors), then p must be a factor of the constant term of f(x), and q must be a factor of the leading coefficient.
Use:
Helps identify possible rational zeros to test using synthetic division or direct substitution. This is often a good place to start.
Statement:
If a polynomial with real coefficients has a complex zero a + bi, then its complex conjugate a - bi is also a zero.
Use:
Helps identify all zeros, including complex ones, if you know one complex zero.
The multiplicity of a zero refers to the number of times a factor appears in the fully factored form of the polynomial. For example, if (x-a) appears twice in the factored form, then 'a' is a zero with a multiplicity of 2.
A one-to-one function is a function where each input maps to a unique output, meaning no two different inputs produce the same output. The horizontal line test is useful, visual way to determine if a function is or isn't one-to-one. Looking at the graph of f(x), if any horizontal line intersects the graph of the function at more than one point, then the function is not one-to-one. (See "Horizontal Line Test and One-to-One Functions" video on YouTube.)
Inverse functions reverse the inputs and outputs of a function. That is, when you put an x value into the original function and get a y value, you can put that y value into the inverse function and get the original x value back. Note that not all functions have inverses; only functions that are one-to-one do, see below for more info. The notation for an inverse function "f-1" (read as "f inverse"), where "f" represents the original function; meaning if you have a function named "f", its inverse would be written as f-1. Keep in mind that the " -1 " does not represent an exponent, but rather signifies the inverse operation.
For example:
original equation:
f(x) = x + 5
replace f(x) with y:
y = x + 5
swap x and y:
x = y + 5
solve for y:
y = x - 5
replace y with f-1(x):
f-1(x) = x - 5
You can verify one function is the inverse of another by using composition:
If f is one-to-one then g is the inverse of f if:
(f ○ g)(x) = x for every x in the domain of g
(g ○ f )(x) = x for every x in the domain of f
Check out the following resources for more support:
An exponential function calculates the exponential growth or decay of data. The basic exponential function is f(x) = ax, where x is the input variable as an exponent, and a is a positive number greater than 1.
Note: Exponential functions are useful for modeling real-world phenomena like populations, radioactive decay, interest rates, and the amount of medicine in the bloodstream!
CHARACTERISTICS OF THE GRAPH f(x) = ax |
| The points (-1, 1/a) and (0,1) and (1,a) are on the graph. |
|
If a is greater than 1, then f is an increasing function. On the other hand, if a is less than 1 but greater than 0, f is a decreasing function. a > 0 → increasing 0 < a < 1 → decreasing |
| The x-axis is the horizontal asymptote. |
| The domain is (-∞, ∞), and the range is (0, ∞) |
Logarithms are a way of writing and working with the exponents. A logarithm is the power to which a number must be raised in order to get some other number For example log3 9= 2, because 3 must be raised to the power of 2 to in order to equal 9.
If a > 0, a ≠ 1, and x > 1 then the logarithmic function with base a is as follows:
Logax = n
Read as "Log base a of x equals n."
Where "a" is the base, "x" is the argument (also called the answer), and "n" is the exponent.
The following table shows several pairs of equivalent statements:
| EXPONENTIAL FORM | LOGARITHMIC FORM |
| 23 = 8 | log28 = 3 |
| 102 = 100 | log10100 = 2 |
| 51 = 5 | log55 = 1 |
| 3-2 = 1/9 | log31/9= -2 |
| 0.5-4 = 16 | log0.516 = -4 |
CHARACTERISTICS OF THE GRAPH f(x) = logax |
| The points (1/a, -1), (1,0) and (a,1) are on the graph. |
|
If a is greater than 1, then f is an increasing function. On the other hand, if a is less than 1 but greater than 0, f is a decreasing function. a > 0 → increasing 0 < a < 1 → decreasing |
| The y-axis is a vertical asymptote. |
| The domain is (0, ∞), and the range is (-∞, ∞) |
Properties of Logarithms
For x > 0, y > 0, a > 0, a ≠ 1, and "r" is any real number:
Theorem on Inverses
For a > 0, a ≠ 1, the following properties hold:
alogax = x (for x > 0) and logaax = x
Check out the following resources for more support:
The two most important bases for logarithms are base 10 and base e. Base 10 logarithms are called common logarithms. The common logarithm of x is written log x where the base is understood to be 10. Basically, is you seem something like log 2 or log 9, it is implied that the base is 10.
log x = log10x
Logarithms with base e are called natural logarithms and are written as ln x (pronounced "L n x" or "El-en x"). They're called natural logarithms because they occur in natural situations that involve growth or decay.
ln x = lnex
Recall that "e" represents a fundamental constant, often referred to as Euler's number, and is approximately equal to 2.71828. It's the limit of the expression (1 + 1/n)n as n approaches infinity. (You'll learn more about limits if you study calculus.)
Many calculators only have log and ln built in, making logarithms with other bases difficult to calculate. However, the change of base theorem is useful for evaluating logarithms where the base is something other than 10 or e. For any positive, real numbers x, a, and b where neither a nor b = 1, the following is true:
For example, to calculate log721 you can use the change of base theorem and compute the following:
Property of Logarithms:
If x > 0, y > 0, a > 0, and a ≠ 1, then the following holds:
x = y is equivalent to logax = logay
Check out the following resources for more support:
In many situations in ecology, biology, economics, and the social sciences, a quantity (y) changes at a rate proportional to the amount present (y0). The amount present at time t is a special function of t called an exponential growth or decay function:
Let y0 be the amount or number present at time t = 0. Then, under certain conditions, the amount y present at any time t is modeled by
y = y0ekt, where k is a constant
The constant k determines the type of function. When k > 0, the function describes growth. When k < 0, the function describes decay.
A system of linear equations is a collection of two or more linear equations involving the same variables. For example 2x + 3y = 15 and x - 2y = 7 are both linear equations with two variables (x and y). When considered together, they form a system of linear equations.
Check out the following resources for more support:
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