Comprehensive Lecture Notes – Exponents, Rational Expressions, Functions & Lines

Exponent Rules and Negative Powers

Any non-zero base raised to a negative power is the reciprocal of the base raised to the corresponding positive power: $a^{-n}=\frac{1}{a^{n}}$ .
Negative exponents applied to products or quotients distribute to every factor: $(ab)^{-n}=a^{-n}b^{-n},\;\;\left(\frac{a}{b}\right)^{-n}=\frac{b^{n}}{a^{n}}$ .
Two legitimate strategies when simplifying an expression such as $(3x^{2}y^{5}z^{2}/5x^{2})^{-3}$ :
- Simplify inside first (e.g.
  • cancel common factors like $x^{2}$ on top and bottom,
  • reduce any numeric fractions), then raise to the power.
- Raise every factor to the power first and simplify afterwards ("flip first, then simplify").
- Both end at the same answer if no algebraic error is made.
Mental-math examples mentioned:
• $3^{2}=9$
• $5^{3}=125$
• $z^{2}\,\text{cubed}=z^{6}$ , $y^{5}\,\text{cubed}=y^{15}$ , etc.

Simplifying Expressions with Negative Exponents (Worked Example)

Start with a fraction containing $x^{3}$ and $x^{2}$ :
• Cancel $x^{2}$ to get a single $x$ in the numerator.
Ignore the constant 5 momentarily, finish all symbolic cancellations, then attach the numeric power:
• After cubing: $3^{3}=27,\;5^{3}=125$ .
If the expression is not a fraction, you can still "flip" by taking the reciprocal of every factor: the reciprocal of 3 is $\frac{1}{3}$ , of 5 is $\frac{1}{5}$ , etc.

Cancellation and Simplification of Rational Expressions

Goal: rewrite a rational expression with as few factors as possible.
Fundamental cancellation rules stated in class:
1. One factor cancels one factor (e.g.
  $\frac{x(x-7)}{(x+1)(x-7)} \to \frac{x}{x+1}$ ).
2. One factor, once isolated, may cancel every term in a product (not in a sum).
Anti-rule (common mistake): You may NOT cancel a single factor sitting in a term of a sum/difference unless the entire numerator & denominator are factored.
Recommended workflow
1. Factor the numerator completely.
2. Factor the denominator completely.
3. Cancel common factors only, never parts of summed expressions.

Factoring Patterns to Remember

Difference of squares: $a^{2}-b^{2}=(a-b)(a+b)$ .
Sum of squares does not factor over the reals: $x^{2}+16$ is irreducible for this course.
Quadratic trinomials:
• If the leading coefficient is 1, look for two numbers that multiply to $c$ and add to $b$ in $x^{2}+bx+c$ .
• Example: $x^{2}-6x-7=(x+1)(x-7)$ because $1\cdot(-7)=-7$ and $1+(-7)=-6$ .

Understanding the Question (Solve vs. Factor vs. Simplify)

Always ask: “What is the problem actually requesting?”
• "Solve for x" ⇒ produce values of $x$ that satisfy the equation, e.g. $x=2, x=-3$ .
• "Factor" ⇒ express the given expression as a product of simpler factors—do not set it equal to zero.
• "Simplify" ⇒ combine like terms, cancel factors, remove negative exponents, etc.
An answer that is unrelated to the requested task is "not even wrong"—it addresses a different question entirely.

Functions: Definitions & Representations

A function is a special relation where every input $x$ has at most one output $y$ .
Three common representations discussed:
1. Set of ordered pairs ⟨ $x,y$ ⟩ (e.g.
  ${(2,5),(3,-7),(6,2)}$).
2. Graph (visual plot of all ordered pairs).
3. Algebraic rule (formula) such as $y=f(x)=x^{2}+4x-2$ .
Vertical Line Test: a graph is a function iff no vertical line intersects it more than once (otherwise one $x$ would have two $y$ ’s).
Example of non-function: a graph that contains both $(0,3)$ and $(0,2)$ fails the test.

Domain and Range

Domain = set of all allowable $x$ values.
Range = corresponding set of $y$ values.
From a graph example given in class:
• Smallest $x$ observed = 1, largest $x$ = 7 → domain $1 \le x < 7$ .
• Smallest $y$ observed = –4, largest $y$ = 3 → range $-4 \le y \le 3$ .

Function Notation and Evaluation

General placeholder idea: $f(\text{input})=\text{rule involving that input}$ .
Examples:
• $f(1)=1^{2}+4(1)-2=3$ .
• $f(z)=z^{2}+4z-2$ (replace every $x$ by $z$ ).
• $f(x^{3})=(x^{3})^{2}+4(x^{3})-2=x^{6}+4x^{3}-2$ .
• $f(0)=-2$ because $0^{2}+4(0)-2=-2$ .

Domain of a Rational Function

Rule given: The domain of $\dfrac{P(x)}{Q(x)}$ is “all real numbers except values that make $Q(x)=0$ .”
Procedure
1. Set the denominator equal to zero.
2. Solve for $x$ .
3. State “all real $x$ except” those solutions.
Do not cancel before deciding domain; canceling might eliminate a factor that actually caused a zero denominator in the original function.
• Example: $\dfrac{4x}{x(x-6)}$ has forbidden $x=0$ and $x=6$ even though $x$ could cancel algebraically—original denominator still goes to zero at those points.

Lines and Slope Basics

Two-point slope formula: $m=\dfrac{y2-y1}{x2-x1}$ .
Interpretations:
• $m>0$ → rising line (up to the right).
• $m<0$ → falling line.
• $m=0$ → horizontal line $y=c$ .
• $m=\text{undefined}$ → vertical line $x=c$ (denominator zero).

Algebraic Forms of a Line

Point–slope: $y-y1 = m(x-x1)$ (often most convenient; instructor’s preference).
Slope–intercept: $y = mx + b$ .
General/standard: $Ax + By = C$ .
All three represent the same geometric object; they are just different descriptions (analogy: “Valerie has an older sister Emily” vs. “Emily has a younger sister Valerie”).

Finding Intercepts & Quick Graphing

X-intercept(s): set $y=0$ , solve for $x$ (result written as either number or point $(x,0)$ ).
Y-intercept: set $x=0$ , solve for $y$ .
For graphing a line quickly: find any two points (commonly the two intercepts), use a straight edge/student ID to draw the line through them.

Parallel and Perpendicular Lines

Parallel lines: equal slopes (provided both are not vertical).
• If both are vertical, they are also parallel even though the slope is "undefined".
Perpendicular lines: slopes are negative reciprocals: $m1 \cdot m2 = -1$ (again, except vertical/horizontal pair where one slope is 0 and the other undefined).
Constructing a line:
1. Extract the slope from the given line.
2. Keep it for a parallel line or take its negative reciprocal for a perpendicular line.
3. Plug the slope and a known point into point–slope form and simplify if required.
  • Example used: Line through $(5,-1)$ parallel to a line of slope $-4$ is $y+1=-4(x-5)$ .

Miscellaneous Instructor Advice & Classroom Notes

Work step-by-step; skipping algebra steps invites errors.
Always verify what the question demands before working.
A correct but irrelevant answer is useless (“not even wrong”).
Quizzes and tests will often disallow calculators for basic arithmetic (e.g.
$25\times5=125$ expected by hand).
Raise thumbs up/thumbs down during class if confused; do not rely on peer signals.
Course pacing is rapid (24 total meetings; by the second class you are ~10 % done).
Class notes and graph paper are provided online and on exams; review them for any rushed material.

The provided notes cover a wide range of essential mathematical concepts, including exponent rules, simplifying algebraic expressions, factoring patterns (like difference of squares and quadratic trinomials), and understanding function definitions, domain, and range. It also delves into lines and linear equations, covering slope, different algebraic forms of a line, and properties of parallel and perpendicular lines. Additionally, it highlights key advice for studying mathematics, such as understanding question types (solve, factor, simplify) and working step-by-step to avoid errors.