FORMULAS - MAT 117

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Unit A: Exponents, Simplifying, Factoring, Solving

Order of Operations with Integers and Exponents:

Evaluate expressions within parentheses
Evaluate terms with exponents
Multiply and divide (from left to right)
Add and subtract (from left to right)

Basic Properties of Exponents:

Product rule: $LaTeX: a^m\cdot a^n=a^{m+n}$
Power rule: $LaTeX: (a^m)^n=a^{m\cdot n}$
Power of a product rule:
Quotient rule: $LaTeX: \displaystyle \frac{a^m}{a^n} = a^{m-n}$

Exponent of Zero:

Negative Exponents:

Whole Number Base

- - Rule 1: $LaTeX: \displaystyle a^{-n}=\frac{1}{a^n}$ (Move $LaTeX: \displaystyle a^{-n}$ to the denominator (bottom) and change $LaTeX: \displaystyle -n$ to $LaTeX: \displaystyle n$ )
  - Rule 2: $LaTeX: \displaystyle \frac{1}{a^{-n}} =a^{n}$ (Move $LaTeX: \displaystyle a^{-n}$ to the numerator (top) and change $LaTeX: \displaystyle -n$ to $LaTeX: \displaystyle n$ )

Positive Fraction Base

$LaTeX: \displaystyle \left( \frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n = \frac{b^n}{a^n}$

Multiplying Binomials: FOIL

First, Outer, Inner, Last

Multiplying Conjugate Binomials:

$LaTeX: \displaystyle (A-B)(A+B) = A^2 - B^2$

Factoring a Perfect Square Trinomial:

- $LaTeX: A^2+2AB+B^2 = \left(A+B\right)^2$
- $LaTeX: A^2-2AB+B^2 = \left(A-B\right)^2$

Finding n^th Roots of Perfect n^th Powers: $LaTeX: x^n = A \hspace{.5em} \Leftrightarrow \hspace{.5em} x = \sqrt[n]{A}$

Converting between Radical and Exponent form:

- $LaTeX: \displaystyle a^{\frac{1}{n}} = a^{1/n} = \sqrt[n]{a}$
- $LaTeX: \displaystyle x^{\frac{m}{n}} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m$

Product Property of Square Roots: (Use to simplify the square root of a whole number greater than 100)

$LaTeX: \sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$

Square Root Multiplication:

$LaTeX: \sqrt{a} \cdot \sqrt{b} = \sqrt{ab},$ for any nonnegative numbers a and b

Scientific Notation: A number written in scientific notation is a number between 1 and 10 multiplied by a power of 10.

Unit B: Solving, Introduction to Functions, Linear Functions, Systems of Equations

Solving an Absolute Value Equation: LaTeX: |A| = B

- If $LaTeX: B \ge 0,$ then or
- If , then has no solution.

Meaning of Inequalities: (Use to write inequality for a real-world situation)

- "A is less than B."
- $LaTeX: A \le B$ "A is less than or equal to B." OR "A is at most B." OR "A is no more than B."

- "A is more than B." OR "A is greater than B."
- $LaTeX: A \ge B$ "A is more than or equal to B." OR "A is greater than or equal to B." OR "A is at least B." OR "A is no less than B."

Solving a Square Root: $LaTeX: ( \sqrt{w})^2 = w$

Classifying Slopes Given Graphs of Lines:

A line that goes upward from left to right has a positive slope.
A line that goes downward from left to right has a negative slope.
A horizontal line goes straight across from left to right and has a slope of zero.
A vertical line goes straight up and down and has an undefined slope.

Slope of a Line: $LaTeX: \displaystyle \text{slope}=\frac{\text{rise}}{\text{run}}$

Given two points (x₁ , y₁) and (x₂ , y₂) on a non-vertical line,

the slope of the line is given by $LaTeX: \displaystyle m=\frac{y_2 - y_1}{x_2-x_1}$

Slope-Intercept Form of a Line:

$LaTeX: \displaystyle y=mx+b,$ where m is the slope and b is the y-intercept

Function: A graph represents a function if and only if no two points on the graph have the same x-coordinate.

Vertical Line Test:

If we are able to draw a vertical line that intersects the graph more than once, then the graph fails the vertical line test. It does not represent a function.
If it is impossible to draw a vertical line that intersects the graph more than once, then the graph passes the vertical line test. It does represent a function.

Domain of a Function: The domain of a function is the set of all inputs that make the function a real number, i.e., all inputs for which it is possible to evaluate the function.

Range of a Function: The range of a function is the set of all outputs of the function.

Average Rate of Change of a Function from x₁to x₂: $LaTeX: \displaystyle \frac{f(x_2) -f(x_1)}{x_2-x_1}, \hspace{1em} x_1 \ne x_2$

Distance, Rate, Time: $LaTeX: \displaystyle \mathrm{distance} = \mathrm{rate} \cdot \mathrm{time}$

Unit C: Graphs of Functions; Composite and Inverse Functions; Quadratic, Polynomial and Rational Functions

The Square Root Property: $LaTeX: \displaystyle A^2 = n \hspace{.5em} \Leftrightarrow \hspace{.5em} A = \sqrt{n} \hspace{.5em} \textbf{or} \hspace{.5em} A = -\sqrt{n}$

Quadratic Formula:

$LaTeX: \displaystyle x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$ are the solution(s) to $LaTeX: \displaystyle ax^2+bx+c = 0$

Complex Roots (Definition of i): $LaTeX: \displaystyle \sqrt{-1} = i$

Graphing a Parabola of the Form $LaTeX: \displaystyle y = ax^2$ :

- If $LaTeX: \displaystyle a > 0$ , the parabola opens upward (like $LaTeX: \cup$ ). The vertex is the lowest point on the graph.
- If $LaTeX: \displaystyle a < 0$ , the parabola opens downward (like $LaTeX: \cap$ ). The vertex is the lowest point on the graph.
- In both cases, the axis of symmetry is the (vertical) line that divides the parabola into two "mirror images." This line goes through the vertex.

Graphing a Parabola of the Form $LaTeX: \displaystyle y = ax^2 +c$ : Vertex is at $LaTeX: \displaystyle (0,c)$

Parabola Vertex Form: $LaTeX: \displaystyle y = a(x-h)^2+k$ , where (h,k) is the vertex

Parabola Standard Form: $LaTeX: \displaystyle y = ax^2+bx+c$ , where $LaTeX: \displaystyle x=-\frac{b}{2a}$ is the axis of symmetry.

Writing a Quadratic Function Given its Zeros:

- If p and q are both zeros of a quadratic function g, then we can write the equation for g as $LaTeX: \displaystyle g(x) = a(x-p)(x-q),$ where a is some nonzero constant
- If p is the only zero of a quadratic function g, then p is a repeated root, and we can write the equation for g as $LaTeX: \displaystyle g(x) = a(x-p)(x-p) = a (x-p)^2$ , where a is some nonzero constant

Difference Quotient of a Function: $LaTeX: \displaystyle \frac{f(x+h) - f(x)}{h}, \hspace{1em} \text{where} \hspace{.3em} h \ne 0$

Finding where a Function is Increasing, Decreasing, or Constant Given the Graph:

- A function f is (strictly) increasing on an interval if, for all a and b in that interval, $LaTeX: \displaystyle a < b$ implies that $LaTeX: \displaystyle f(a) < f(b)$ . (Graph goes up from left to right.)
- A function f is (strictly) decreasing on an interval if, for all a and b in that interval, $LaTeX: \displaystyle a < b$ implies that $LaTeX: \displaystyle f(a) > f(b)$ . (Graph goes down from left to right.)
- A function f is constant on an interval if, for all a and b in that interval, $LaTeX: \displaystyle f(a) = f(b)$ . (Graph is horizontal.)

Graphing an Absolute Value Equation (Shifting up/down versus left/right):

For $LaTeX: \displaystyle y = a|x|+b$ , the vertex occurs at $LaTeX: \displaystyle x=0$
For $LaTeX: \displaystyle y = a|x -b|$ , the vertex occurs at the x-value that makes $LaTeX: \displaystyle x-b=0$

Symmetry of Graphs:

A graph is symmetric with respect to the x-axis if for every point (x,y) on the graph, the point (x, -y) is also on the graph.
A graph is symmetric with respect to the y-axis if for every point (x,y) on the graph, the point (-x, y) is also on the graph. Also called an even function.
A graph is symmetric with respect to the origin if for every point (x,y) on the graph, the point (-x, -y) is also on the graph. Also called an odd function.

Even and Odd Functions:

Even function: for all x in the domain. The graph of an even function is symmetric with respect to the y-axis.
Odd function: for all x in the domain. The graph of an odd function is symmetric with respect to the origin.

Translating the Graph of a Parabola (One Step):

Horizontal Translations: LaTeX: (c > 0)

- To graph , we shift the graph of $LaTeX: \displaystyle y=x^2$ to the right c units.
- To graph , we shift the graph of $LaTeX: \displaystyle y=x^2$ to the left c units.

Vertical Translations: LaTeX: (c > 0)

- To graph , we shift the graph of $LaTeX: \displaystyle y=x^2$ upward c units.
- To graph , we shift the graph of $LaTeX: \displaystyle y=x^2$ downward c units.

Translating the Graph of a Parabola (Two Steps):

To graph , we shift the graph of $LaTeX: \displaystyle y=x^2$ to the right a units and upward b units.
To graph , we shift the graph of $LaTeX: \displaystyle y=x^2$ to the right a units and downward b units.
To graph , we shift the graph of $LaTeX: \displaystyle y=x^2$ to the left a units and upward b units.
To graph , we shift the graph of $LaTeX: \displaystyle y=x^2$ to the left a units and downward b units.

Translating the Graph of an Absolute Value Function (One Step):

Horizontal Translations: LaTeX: (c > 0)

- To graph , we shift the graph of $LaTeX: \displaystyle y=|x|$ to the right c units.
- To graph , we shift the graph of $LaTeX: \displaystyle y=|x|$ to the left c units.

Vertical Translations: LaTeX: (c > 0)

- To graph , we shift the graph of $LaTeX: \displaystyle y=|x|$ upward c units.
- To graph , we shift the graph of $LaTeX: \displaystyle y=|x|$ downward c units.

Translating the Graph of an Absolute Value Function (Two Steps):

To graph , we shift the graph of $LaTeX: \displaystyle y=|x|$ to the right a units and upward b units.
To graph , we shift the graph of $LaTeX: \displaystyle y=|x|$ to the right a units and downward b units.
To graph , we shift the graph of $LaTeX: \displaystyle y=|x|$ to the left a units and upward b units.
To graph , we shift the graph of $LaTeX: \displaystyle y=|x|$ to the left a units and downward b units.

Translating the Graph of a Function (One Step):

Horizontal translations: LaTeX: (c > 0)

- To graph , we shift the graph of f(x) to the right c units.
- To graph , we shift the graph of f(x) to the left c units.

Vertical translations: LaTeX: (c > 0)

- To graph , we shift the graph of f(x) upward c units.
- To graph , we shift the graph of f(x) downward c units.

Translating the Graph of a Function (Two Steps):

To graph , we shift the graph of $LaTeX: \displaystyle y=f(x)$ to the right a units and upward b units.
To graph , we shift the graph of $LaTeX: \displaystyle y=f(x)$ to the right a units and downward b units.
To graph , we shift the graph of $LaTeX: \displaystyle y=f(x)$ to the left a units and upward b units.
To graph , we shift the graph of $LaTeX: \displaystyle y=f(x)$ to the left a units and downward b units.

Combining functions, f and g: (A is domain of f, B is domain of g)

- - Domain: $LaTeX: A \cap B$
  - Domain: $LaTeX: A \cap B$
  - $LaTeX: (f\cdot g)(x)=f(x) \cdot g(x)$ Domain: $LaTeX: A \cap B$
  - $LaTeX: \displaystyle \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$ Domain: $LaTeX: A \cap B - \{x | g(x) = 0\}$

Composition of two functions, f and g: $LaTeX: (f \circ g)(x) = f(g(x))$

One-to-one and the Horizontal Line Test: (Use to determine if a function has an inverse without having to restrict its domain.)

One-to-one: A function is one-to-one if and only if no two points on its graph have the same y-coordinate.

Horizontal line test:

- If we are able to draw a horizontal line that intersects the graph more than once, then the graph fails the horizontal line test. The function is not one-to-one.
- If it is impossible to draw a horizontal line that intersects the graph more than once, then the graph passes the horizontal line test. The function is one-to-one.

Inverse functions:

The functions, f and g, are inverses of one each other if and only if both of the following conditions are true:

Condition 1: for all x in the domain of g
Condition 2: for all x in the domain of f

Domain and Range of a function, , and its inverse, $LaTeX: f^{-1}$ :

The domain of is the range of $LaTeX: f^{-1}$ .
The domain of $LaTeX: f^{-1}$ is the range of .

Identifying Polynomial Functions:

A polynomial function in one variable, x, is a function that can be written in the following form $LaTeX: P(x) = a_nx^n + a_{n-1}x^{n-1} + \dots a_1x+a_0$ . The coefficients $LaTeX: a_n, a_{n-1}, \dots a_1, a_0$ are real numbers, and the degree n is a whole number.

Finding zeros and their multiplicities of factored polynomials:

The number c is a zero of multiplicity k of a polynomial function if and only if k is the largest positive integer such that LaTeX: (x-c)^k is a factor of the polynomial.

Rational function vertical asymptotes: After the function has been written in its simplest form (no common factors in the numerator and denominator), then the vertical asymptotes are the values of x (or the input variable) that makes the denominator equal to zero.

Rational function horizontal asymptotes: To find the horizontal asymptotes (if any), compare the degree n of the numerator with the degree m of the denominator.

If n < m, the horizontal asymptote is y = 0.
If n > m, there is no horizontal asymptote.
If n = m, then the horizontal asymptote is given by $LaTeX: \displaystyle y=\frac{\text{leading coefficient of the numerator}}{\text{leading coefficient of the denominator}}$

Unit D: Exponential and Logarithmic Functions and Applications

Converting between logarithms and exponential functions:

Logarithms base a: $LaTeX: \displaystyle \log_ac = b$ if and only if $LaTeX: \displaystyle a^b = c$
Natural logarithms (base e): $LaTeX: \displaystyle \ln c = b$ if and only if $LaTeX: \displaystyle e^b = c$

Change of base of logarithm from base a to base c: (a, b, and c positive real numbers with $LaTeX: \displaystyle a \ne 1$ and $LaTeX: \displaystyle c \ne 1$ )

$LaTeX: \displaystyle \log_a b = \frac{\log_c b}{\log_c a}$

Basic properties of logarithms:

Product: $LaTeX: \displaystyle \log_a(MN) = \log_aM + \log_aN$
Quotient: $LaTeX: \displaystyle \log_a\left(\frac{M}{N}\right) = \log_aM - \log_aN$
Power: $LaTeX: \displaystyle \log_aM^p = p\log_aM$

Half-life or doubling time: $LaTeX: \displaystyle y = y_0e^{rt}$ , where

$LaTeX: \displaystyle y$ = the amount at time t, or the final amount
$LaTeX: \displaystyle y_0$ = the amount at time t = 0, or the initial amount
e = a particular constant (often referred to as Euler's constant)
$LaTeX: \displaystyle r$ = the relative rate of growth (r > 0) or decay (r < 0)
$LaTeX: \displaystyle t$ = time t

Compound Interest Continuous $LaTeX: \displaystyle A = Pe^{rt}$ , where

$LaTeX: \displaystyle A$ = the amount at time t, or the final amount
$LaTeX: \displaystyle P$ = the amount at time t = 0, or the initial/principal amount
e = a particular constant (often referred to as Euler's constant)
$LaTeX: \displaystyle r$ = the annual interest rate
$LaTeX: \displaystyle t$ = time t measured in years

Compound Interest (Not Continuous) $LaTeX: \displaystyle A = P\left(1 + \frac{r}{n}\right)^{nt}$ , where

$LaTeX: \displaystyle A$ = the amount at time t, or the final amount
$LaTeX: \displaystyle P$ = the amount at time t = 0, or the initial/principal amount
$LaTeX: \displaystyle r$ = the annual interest rate
$LaTeX: \displaystyle n$ = the number of times interest is compounded each year
$LaTeX: \displaystyle t$ = time t measured in years