Parallel & Perpendicular Lines; Functions, Domain, and Vertical Line Test

Parallel & Perpendicular Lines

• Problem set-up (student’s Wi-Fi lagged; instructor reads numbers aloud)
  • Find equation of a line parallel to $2y+3x=5$ through point $(-1,7)$.
  • Find equation of a line perpendicular to $3y+4x=7$ through point $(-2,3)$.

Recap: Slope–Intercept Form

• General form: $y=mx+b$
  • $m$ = slope (rise/run)
  • $b$ = y-intercept (point $(0,b)$)
• To write equation you need one point + slope (intercept can be solved for later).

Example 1 – Parallel Line

1. Identify slope of given line.
   • Solve $2y+3x=5$ for $y$:
   \begin{aligned}2y &= -3x+5\
   y &= -\frac{3}{2}x + \frac{5}{2}\end{aligned}
   • Slope $m=-\frac{3}{2}$.
2. Parallel ⇒ same slope.
   Start: $y=-\tfrac{3}{2}x+b$.
3. Plug point $(-1,7)$ to solve $b$:
   $7 = -\frac{3}{2}(-1)+b \;\Rightarrow\; 7 = \frac{3}{2}+b$
   $b = 7-\frac{3}{2}=\frac{11}{2}$.
4. Final equation: $\boxed{y=-\frac{3}{2}x+\frac{11}{2}}$.

Example 2 – Perpendicular Line

1. Slope of $3y+4x=7$:
   \begin{aligned}3y &= -4x+7\
   y &= -\frac{4}{3}x+\frac{7}{3}\end{aligned}
   • Original slope $m=-\frac{4}{3}$.
2. Perpendicular ⇒ negative reciprocal (change sign & flip):
   $m_{\perp}=+\frac{3}{4}$.
3. Equation with unknown intercept: $y=\frac{3}{4}x+b$.
4. Plug point $(-2,3)$:
   $3 = \frac{3}{4}(-2)+b = -\frac{6}{4}+b=-\frac{3}{2}+b$
   $b = 3+\frac{3}{2}=\frac{9}{2}$.
5. Final equation: $\boxed{y=\frac{3}{4}x+\frac{9}{2}}$.

Key Take-Aways

• Parallel lines → identical slope.
• Perpendicular lines → slopes are opposite reciprocals.
• Slope determines direction; intercept adjusts vertical position.

Introduction to Functions

Formal Definition

• A function $f$ is a rule (mapping) that assigns each element $x$ in a domain $A$ to exactly one element $y$ in a range $B$.
  • Notation: $f:A\to B\,,\; f(x)=y$.
• “Exactly one” is crucial—if an $x$ produces two different $y$’s, relation is not a function.

Mapping Diagram Illustration

• Valid function:
  $x<em>1\to y</em>1\, ,\; x<em>2\to y</em>2\, ,\; x<em>3\to y</em>3\, ,\; x<em>4\to y</em>1$
  (multiple $x$ can share a $y$).
• Invalid: $x<em>2$ mapping to $y</em>2$ and $y_3$ (one $x$ → two $y$’s).

Point-Set Examples

1. ${(-1,\tfrac12),(2,3),(5,7),(\tfrac14,\sqrt2)}$ → each $x$ unique ⇒ function,
   • Domain ${-1,2,5,\tfrac14}$, Range ${\tfrac12,3,7,\sqrt2}$.
2. ${(0,0),(3,9),(\sqrt2,4),(-3,9)}$ → still a function (shared $y=9$ ok).
3. ${(-\tfrac13,5),(2,7),(-3,4),(2,-1),(5,8)}$ → not a function (same $x=2$ → two $y$’s).

Vertical Line Test (Graphical Criterion)

• Draw any vertical line $x=c$:
  • If it meets the curve at most once, curve represents a function.
  • If it intersects more than once, some $x$ has multiple $y$’s ⇒ not a function.
• The test is nothing more than visualizing the formal “exactly-one-y” rule.

Function Notation & Evaluation

• Symbols: $f(x),\; g(x),\; k(x)$ etc. (variable names may vary).
• Input = $x$, Output = $f(x)$ (often called $y$).

Worked Example

Given $f(x)=-x^2+5x+3$, compute:

• $f(2)$: $-(2)^2+5\cdot2+3 = -4+10+3 = 9$.
• $f(-3)$: $-(-3)^2+5(-3)+3 = -9-15+3 = -21$.
• $f(-\tfrac12)$:
  $-\left(-\tfrac12\right)^2 + 5\left(-\tfrac12\right)+3 = -\tfrac14 - \tfrac52 + 3 = \tfrac14$.
  • Demonstrates fractions handled same way; still single output.

Mapping Interpretation

• $2\mapsto9\,,\; -3\mapsto-21\,,\; -\tfrac12\mapsto\tfrac14$.
• Assignments here are rule-based, not arbitrary.

Domain & Range Fundamentals

Formal Domain Definition

“Domain of $f$” = set of **all $x$-values for which $f(x)$ is *defined* (produces a real answer).”

Polynomials

• Contain only powers, sums, differences ⇒ no restrictions.
• Example above: domain $(-\infty,\infty)$ (all real numbers).

Rational Functions

• Denominator ≠ 0.
• Example: $f(x)=\dfrac{1}{x+1}$
  • Undefined at $x=-1$ (division by zero).
  • Domain: all reals except $-1$.
  – Plain English: “all real numbers except $-1$”.
  – Set-builder: ${x\in\mathbb R\mid x\neq-1}$.
  – Interval: $(-\infty,-1)\cup(-1,\infty)$.

Notation Recap

• “Union” symbol $\cup$ joins two intervals/sets.
• Parentheses ( ) exclude an endpoint; brackets [ ] include it.

Quick Checklist for Finding Domain

1. Fractions: set denominator $\neq 0$.
2. Even roots: radicand $\ge 0$ (for real-valued functions).
3. Logarithms: argument >0.
4. Other special operations (e.g.
   tangent) → watch asymptotes.

Practical Insights & Connections

• Parallel/perpendicular concepts tie directly into slope (linear algebra foundation).
• Understanding “exactly one output” prepares students for advanced topics (inverse functions, one-to-one, calculus limits).
• Vertical line test offers quick graphical diagnostic used in pre-calculus and beyond.
• Domain analysis cultivates habit of checking where a formula actually makes sense—vital for solving equations, graphing, and integration limits.
• Set, interval, and function notation create a precise mathematical language, replacing long English descriptions and enabling compact proofs.

Key Formulas & Summary Table

• Slope–Intercept: $y=mx+b$.
• Parallel slopes: $m<em>1=m</em>2$.
• Perpendicular slopes: $m<em>1\cdot m</em>2=-1$ or $m<em>2=-\dfrac{1}{m</em>1}$.
• Domain (polynomial): $(-\infty,\infty)$.
• Domain (rational): exclude roots of denominator.
• Vertical Line Test ⇔ Function criterion.

Ethical / Practical Notes

• Instructor adapts pace (re-reading data) for students with tech issues—emphasizes accessibility.
• Explicit clarification of notation (union, exclusion) prevents miscommunication in collaborative work.
• Encourages students to “create their own functions,” reinforcing creative mathematical thinking.