Precalculus – VANG, Function Notation & Calculator Review

VANG – Four Representations

• ALWAYS describe functions Verbally, Algebraically, Numerically, Graphically.
• Skill goal: move smoothly among the four forms.

Linear Model Example (Bob’s Cakes)

• Verbal: “Bob starts with 3 cakes and bakes 4 cakes every 2 h.”
• Algebraic: $y = 3 + 2x$ (slope $2\,\text{cakes/h}$, intercept $3$)
• Numeric: table uses $x$ = time (h), $y$ = cakes → ${(0,3),(1,5),(2,7),\dots}$
• Graphic: plot intercept $\,(0,3)$; rise $2$, run $1$ to sketch line.

Function Notation Basics

• $f(a)$ means “output when $x=a$; replace every $x$ by $a$.”
• Evaluation: if $f(x)=9-4x$ then
  • $f(-3)=21$
  • $f(x)=10$ ⇒ $x=-\tfrac14$
  • $f(h)=9-4h$, $f(x+2)= -4x+1$ after simplification.

Quadratic / Circle Area Example

• Area as a function of radius: $A(r)=\pi r^{2}$.
• Table for $r=-2,-1,0,1,2$ gives ${4\pi,\pi,0,\pi,4\pi}$.
• Graph is a parabola opening up (symmetry about $r=0$).

Calculator Skills (TI-84 style)

• Enter equations under Y= (always uses $x, y$ symbols).
• Standard window: Zoom 6 ( $x,y\in[-10,10]$ ). Adjust via WINDOW.
• WINDOW choices:
  • $X<em>{min},X</em>{max}$ = domain (e.g. $0\to7$ days).
  • $Y<em>{min},Y</em>{max}$ = reasonable output range.
  • $X<em>{Scl},Y</em>{Scl}$ = tick-mark spacing.
• Trace: gives coordinates; type a value then ENTER to jump to $x$ of interest (e.g. trace 3 gives $c(3)=35$).
• Finding specific $y$ values: approximate by moving cursor or use CALC menu later.
• TABLE (2nd GRAPH): shows numeric pairs.
  • TABLE SET (2nd WINDOW):
  – TblStart: first $x$ shown.
  – ΔTbl: step size.
  – Indpnt: ASK lets you type any $x$ to retrieve $y$.
  • Do NOT switch Depend to ASK.

Intercepts & Extrema via Graph

• $y$-intercept: evaluate $f(0)$ (e.g. $50$ cards).
• Minimum / maximum: visually locate lowest/highest point; refine later with CALC-min/max.

Key Takeaways

• Master translating among words, equations, tables, and graphs (VANG).
• Treat function notation as systematic substitution.
• Use calculator windows that match the problem’s domain & range.
• Trace and Table provide quick numeric checks without manual substitution.