Advanced Algebra 2 - Math Notes - Unit 1
1.1.1- Functions
A relationship between inputs and outputs is a function if there is no more than one output for each input. Functions
are often written as y = some expression involving x, where x is the input and y is the output. The following is an
example of a function.
In the example above the value of y depends on x, so y is also called the dependent variable and x is called
the independent variable.
Another way to write a function is with the notation "f(x) =" instead of “y =”. The function named “f” has
output f(x). The input is x.
In the example at right, f(5) = 9. The input is 5 and the output is 9. You read this as, “f of 5
equals 9.”
The set of all inputs for which there is an output is called the domain. The set of all possible
outputs is called therange. In the example above, notice that you can input any x-value into
the equation and get an output. The domain of this function is “all real numbers” because any
number can be an input. The outputs are all greater than or equal to zero, so the range is y ≥ 0.
x2 + y2 = 1 is not a function because there are two y-values
(outputs) for some x-values, as shown below.
1.1.2 - Linear Equations
A linear equation is an equation that forms a line when it is graphed. This type of equation may be written in several
different forms. Although these forms look different, they are equivalent; that is, their graphs are all the same line.
Standard Form: An equation in ax + by = c form, such as −6x + 3y = −18.
Slope-Intercept Form: An equation in y = mx + b form, such as y = 2x −6.
You can find the slope (also known as the growth factor) and the yintercept of a line in y = mx + bform quickly. For the equation y = 2x − 6, the
slope is 2, while the y −intercept is (0, −6).
1.1.3- Domain and Range
The set of possible values for the input of a function is called the domain of the function. This set consists of every input
value for xfor which the function is defined.
The range of a function is the set of possible values of the output. This set contains every y-value that the function can
generate.
Domain and range are often written with inequality notation as shown in the examples below.
The symbols −∞ and ∞ represents positive and negative infinity. They mean that the domain goes on without ending in
the positive or negative direction. Infinity is not a number; it is a concept.
If the domain is any number between and including −2 and 7:
−2 ≤ x ≤ 7
If the range is any number greater than but excluding 4:
y > 4 or 4 < y < ∞
If the domain is all numbers except for −3:
x ≠ −3
If the domain is all real numbers:
−∞ < x < ∞
1.1.4 - Solving a Quadratic Equation
In a previous course, you learned how to solve quadratic equations (equations that can be written in the form
ax2 + bx + c = 0 ). Review two methods for solving quadratic equations below.
Some quadratic equations can be solved by factoring and then using the Zero Product Property. For example, the
quadratic equation x2 − 3x − 10 = 0 can be written by factoring as (x − 5)(x + 2) = 0. The Zero Product Property states that
if ab = 0, then a = 0 or b = 0. So if (x − 5)(x + 2) = 0, then (x − 5) = 0 or (x + 2) = 0. Therefore, x = 5 or x = −2. Another
method for solving quadratic equations is using the Quadratic Formula. This method is particularly helpful for solving
quadratic equations that are difficult or impossible to factor. Before using the Quadratic Formula, the quadratic
equation you want to solve must be in standard form (that is, written as ax2 + bx + c = 0 ). In this form, a is the
coefficient of the x2-term, b is the coefficient of the x-term, and c is the constant term. The Quadratic Formula is stated
below.
This formula gives two possible solutions for x. The two solutions are shown by the "±" symbol. This symbol (read as
"plus or minus") is shorthand notation that tells you to evaluate the expression twice: once using addition and once
using subtraction. Therefore, Quadratic Formula problems usually must be simplified twice to give:
Of course if
√𝑏 2
or
− 4𝑎𝑐 equals zero, you will get the same result both times.
To solve x2 − 3x − 10 = 0 using the Quadratic Formula, substitute a = 1, b = −3, and c = −10 into the formula, as shown
below, then simplify.
x = 5 or x = −2
1.2.1 - Triangle Trigonometry
There are three trigonometric ratios you can use to solve for the missing side lengths and angle measurements in any
right triangle: tangent, sine, and cosine.
In the triangle below, when the sides are described relative to the angle θ (the Greek letter “theta”), the opposite leg
is y and the adjacent leg is x. The hypotenuse is h regardless of which acute angle is used.
In general, for any uniquely determined triangle, missing sides and angles can be determined by using the Law of
Sines or the Law of Cosines.
1.2.2- 1.2.4 Graphs with Asymptotes
A mathematically clear and complete definition of an asymptote requires some ideas from calculus, but some examples
of graphs with asymptotes should help you recognize them when they occur. In the following examples, the dotted lines
are the asymptotes, and the equations of the asymptotes are given. In the two lower graphs, the y-axis, x = 0 , is also an
asymptote.
As you can see in the examples above, asymptotes can be diagonal lines or even curves. However, in this course,
asymptotes will almost always be horizontal or vertical lines. The graph of a function has a horizontal asymptote if as
you trace along the graph out to the left or right (that is, as you choose x-coordinates farther and farther away from
zero, either toward infinity or toward negative infinity), the distance between the graph of the function and the
asymptote gets closer to zero.
A graph has a vertical asymptote if, as you choose x coordinates closer and closer to a certain value, from either the left
or right (or both), the y coordinate gets farther away from zero, either toward infinity or toward negative infinity.