Summary of "TUDO de FUNÇÃO do PRIMEIRO GRAU pro ENEM 2026 (didática insana e muitas questões resolvidas)"

Overall theme

A deep, example-driven review of first-degree (linear) functions aimed at ENEM preparation. Pedro Assad presents linear functions as a practical problem‑solving tool: model real contexts with y = a·x + b, interpret a and b, build tables, plot graphs, and solve forward (given x → find y) and inverse (given y → find x) problems. The lesson repeatedly emphasizes thinking in terms of input → process → output.

Think in terms of input → process → output.

Core concepts taught

Function as a rule mapping input → output (the terms “input” and “output” are used to aid intuition).
Generic linear function: y = a·x + b (also written f(x) = a·x + b).
- a — slope (coefficient angular, rate, price per unit, Δy/Δx).
- b — independent term (linear coefficient, fixed fee, y‑intercept, value when x = 0).
Interpretation of units and context: x is the varying quantity (distance, hours, area, minutes, etc.); y (or f(x), P) is the total/cost/result.
Many real costs split into two components: a fixed component (b) + a variable component proportional to x (a·x).
- Examples: taxi fare (entry fee + per km), painter (flat call + per m²), concrete pumping (start fee + per m³), rental car (daily fee + per km), worker hire (hourly rate + transport), phone plans (monthly package + per-minute overage).
Slope as marginal change: a = Δy / Δx (how much y changes when x increases by one unit).
Finding the function from two known points:
- Substitute each (x,y) into y = a·x + b to build two linear equations, then solve for a and b (substitution or elimination).
- Quick method: compute a = Δy/Δx first, then find b via b = y − a·x.
Graphing linear functions:
- Cartesian plane: horizontal axis = x, vertical axis = y.
- Two distinct points determine the unique line.
- The line crosses the y‑axis at b (value when x = 0).
Inverse use (find x given y): x = (y − b) / a.
Units and scaling: you may choose different units for x (e.g., blocks of 5 years); the math is the same but the interpretation of a changes.
Practical test-taking strategy: translate story/context into y = a·x + b quickly; test multiple‑choice alternatives by plugging plausible x values; use Δy/Δx to get slope when two output values are given; look for y when x = 0 to identify b.

Methodology — step-by-step instructions

Translate the context into the linear model
- Identify the varying quantity and call it x (distance, hours, area, minutes, volume, etc.).
- Identify what you must calculate (total price, revenue, total doctors, etc.) and call it y (or f(x), P, V).
- Find the fixed part (b): what is charged/obtained even if x = 0?
- Find the per‑unit rate (a): how much y increases per one‑unit increase in x.
- Write y = a·x + b.
Compute a and b from data (two‑point method)
- If two points (x1,y1) and (x2,y2) are given:
  - Compute a = (y2 − y1) / (x2 − x1).
  - Compute b = y1 − a·x1 (or b = y2 − a·x2).
- Alternate: set up the two equations y1 = a·x1 + b and y2 = a·x2 + b and solve by elimination or substitution.
Find outputs for given inputs
- Substitute x into y = a·x + b and compute y.
- For large multiplications, separate into a·x and b for clarity.
Find input x when given output y
- Rearrange: x = (y − b) / a.
- Subtract b from both sides, then divide by a.
Graphing a linear function
- Choose an appropriate scale for the x and y axes.
- Plot at least two correct points (often from a table).
- Draw the straight line through the points.
- Read off: y‑intercept = b; slope = rise/run between two plotted points.
Quick test shortcuts and checks
- y‑intercept test: evaluate y when x = 0 to identify b.
- Slope test: compute Δy/Δx between two provided points instead of solving the full system.
- Option testing: plug the same test x into answer choices (in MCQs) — prefer values that simplify arithmetic (like x = 0).
- Unit‑awareness: check that units of a and b make sense (currency per km, fixed currency, number per year, etc.).
- Use elimination by subtracting equations to eliminate b quickly.
Decision/comparison between alternatives (e.g., phone plans or quotes)
- Compute total cost for the exact usage profile under each plan: total = fixed fee + (overage minutes·overage rate) + (minutes to other carriers·other rate).
- Discard plans that are obviously worse (higher fixed cost with no benefit for the given usage).
- Compare computed totals to pick the minimum cost.

Examples and problem contexts covered

Taxi fare: entry fee + price per km → example y = 2x + 5; includes table of values and graph.
Oven/cake: input = ingredients → output = cake (illustrative input/output).
General revenue: number sold × price per item → y = a·x (case b = 0).
Painter pricing: flat call fee + per m² → find a, b from provided points and compute price for 150 m².
Worker hire: hourly rate × hours + fixed transport cost → compute totals & choose cheapest.
Car rental: daily base fee + per‑km rate → model and select expression.
Concrete pumping: fixed pump fee + per m³ cost → example y = 250x + 500.
Phone plans: monthly package V with M minutes + overage rates T1 (same operator), T2 (other operators) → compute costs and compare plans.
Store sales over a month: linear decline in sales; use model to project remaining days and compute how much more is needed to meet monthly target.
Number of doctors: dataset (years vs doctors) modeled as linear growth; compute slope from differences and project to 2040.

Algebra techniques emphasized

Substitute points into y = a·x + b to set up equations.
Solve systems via:
- Substitution (isolate a or b and replace).
- Elimination (subtract equations to cancel b).
Use the distributive property when substituting expressions.
Use Δy/Δx as a faster route to the slope when two outputs are given.

Common pitfalls warned about

Don’t multiply the whole observed total value (which already includes b) when scaling — separate the fixed part b from the variable part a·x before scaling.
Distinguish “price for x units” from “price increase caused by adding x units” — the latter isolates a.
Always check units: mixing up “per km” vs “per day” multipliers leads to wrong expressions (e.g., don’t multiply a daily fixed fee by kilometers).
Use x = 0 correctly to identify b (y‑intercept); don’t treat time/year labels as x without resetting a starting zero point.

Takeaways and exam guidance

Mastering linear functions (input/output thinking, modeling, slope/intercept, graphing) can reliably secure several correct ENEM math questions.
Active practice is stressed: build many examples, test multiple‑choice alternatives, and internalize the input → output mental model.