Linear Functions in Four Forms
Overview
This is a self-contained, low-tech substitute packet in which students practice linear functions. Working alone with a pencil (a basic or graphing calculator is allowed but not required), students evaluate a function in function notation f(x) and find slope from two points, study a worked example that shows one function as a table, a graph, and the equation f(x) = mx + b, and then apply the ideas: matching the four forms, interpreting slope and y-intercept in context, writing an equation from a context, solving f(x) = k, and fixing a mis-found slope. On a block schedule they also model a draining tank and find its x-intercept. They finish by justifying which of two plans costs less with a claim, evidence, and reasoning. A key idea throughout is telling proportional lines (through the origin) apart from non-proportional lines (with a starting amount).
At a glance
Course: Algebra I (Grades 9–12)
Subject: Mathematics
Time: about 50 minutes on a standard period, or about 90 minutes on a block period (block adds an extra applied modeling task)
Materials: printed packet and a pencil; a basic or graphing calculator is allowed but not required (no computer or internet)
Work mode: independent
Standards (provisional): 19 TAC §111.39 (Algebra I) — writing linear equations in slope-intercept form from tables, graphs, and verbal descriptions; slope as a rate of change; interpreting slope and intercepts in context; and solving linear equations including f(x) = k. Provisional — pending educator verification against the current official TAC source. Standards are paraphrased, not quoted; this packet is not claimed to be "aligned to the TEKS" until reviewed.
Accessible version of the student activity
The full student activity is reproduced below in plain, screen-reader-friendly HTML. It reflows on phones and at 200% zoom. Write your answers on the printed packet, and show your work.
Start (6 minutes) — Retrieval warm-up
- Evaluate a function: f(x) = 2x + 3. Find f(4). Show the multiplication and addition. (f(4) means the output when x = 4; it is not f times 4.)
- Find slope from two points: a line passes through (1, 5) and (4, 11). Find the slope m using m = (y₂ − y₁) ÷ (x₂ − x₁).
Build (8–12 minutes) — Study the math
A linear function graphs as a straight line and changes by the same amount for each equal step of x. In function notation, f(x) is the output at input x, so f(4) is the output when x = 4 (not multiplication). The slope (rate of change) is m = rise ÷ run = (y₂ − y₁) ÷ (x₂ − x₁). The y-intercept b is the output when x = 0 — the point (0, b). The x-intercept is where the output is 0 (solve f(x) = 0). Together slope and y-intercept give the slope-intercept form: y = mx + b, or in function notation f(x) = mx + b (y and f(x) both name the output). To solve f(x) = k, replace f(x) with k and solve: mx + b = k → mx = k − b → x = (k − b) ÷ m. The same function can appear as a table (a fixed step in the output for each fixed step in x; the x = 0 row gives b), a graph (a straight line crossing at (0, b)), an equation (f(x) = mx + b), or a context (a starting amount b plus a steady rate m per one). A linear function is proportional only when b = 0 — the line passes through the origin (0, 0). If b is not 0, the line is linear but not proportional. Remember: evaluating starts from x and finds the output; solving starts from the output and finds x.
Worked example — one function, four forms
A tutoring service costs $3 to book (a fee) plus $2 per hour. Let x = hours and f(x) = total cost in dollars. Because there is a starting amount of 3, this is linear but NOT proportional. The slope is m = 2 dollars per hour and the y-intercept is b = 3 dollars, so the equation is f(x) = 2x + 3. Checking the slope from rows (1, 5) and (3, 9): m = (9 − 5) ÷ (3 − 1) = 2. Evaluating: f(4) = 2(4) + 3 = 11. Solving f(x) = 15: 2x + 3 = 15 → 2x = 12 → x = 6. The line crosses the y-axis at (0, 3), not the origin.
| Hours x | Cost f(x) (dollars) | Change in output |
|---|---|---|
| 0 | 3 | — (start) |
| 1 | 5 | +2 |
| 2 | 7 | +2 |
| 3 | 9 | +2 |
| 4 | 11 | +2 |
- Using f(x) = 2x + 3, find f(7) (the cost at 7 hours). Show your work.
Apply (20–26 minutes) — Use what you know
Table B (phone-repair cost g(x)): 0 parts cost $25; 2 parts cost $55; 4 parts cost $85; 6 parts cost $115. Context: a shop charges a one-time $25 diagnostic fee plus $15 for each part.
| Parts x | Total cost g(x) (dollars) |
|---|---|
| 0 | 25 |
| 2 | 55 |
| 4 | 85 |
| 6 | 115 |
- Match the four forms using Table B and the context:
- 4a. Find the slope m (dollars per part); show m = (change in output) ÷ (change in x) using two rows.
- 4b. Find the y-intercept b (cost at 0 parts), then write the equation in function notation, g(x) = mx + b.
- 4c. Interpret in context: what does the slope mean, and what does the y-intercept mean (use "per part" and "diagnostic fee")?
- 4d. Solve g(x) = 145: a repair cost $145 total; how many parts were used? Set up and solve for x, showing every step.
- 4e. Is this function proportional or not proportional? Explain using the y-intercept.
- Write the equation from a context: a gym membership has a $40 sign-up fee plus $12 per month; x = months, h(x) = total cost.
- 5a. Identify m and b, then write h(x) = mx + b.
- 5b. Find h(9), the total cost after 9 months; show your work.
- Error analysis (a mis-found slope): a student used rows (2, 55) and (4, 85) and wrote "m = run ÷ rise = (4 − 2) ÷ (85 − 55) = 2 ÷ 30 ≈ 0.07." Find the mistake, explain it, and give the correct slope.
Block only (~12–15 minutes) — Extra applied task
- Real-world modeling: a tank starts with 60 gallons and drains 4 gallons per minute;
x = minutes, V(x) = gallons left.
- 7a. Write V(x) = mx + b (the rate is negative because the water is decreasing).
- 7b. Find V(6) — gallons left after 6 minutes; show your work.
- 7c. Solve V(x) = 0 (the x-intercept): after how many minutes is the tank empty? Show your steps, then interpret the x-intercept in the story.
Explain (6–10 minutes) — Justify your strategy (Claim, Evidence, Reasoning)
Question 8. Two streaming services: Plan A is f(x) = 3x + 8 ($8 base charge plus $3 per premium channel); Plan B is g(x) = 5x ($5 per channel, no base charge). For 3 premium channels, which plan costs less, and why? Also state which plan is proportional. Write a Claim (your answer), Evidence (the numbers you calculated), and Reasoning (why your strategy works, using slope and y-intercept).
Sentence stems you may use: "For 3 channels, Plan ___ costs less because…"; "I found Plan A's cost by… and Plan B's cost by…"; "Plan B is proportional because its y-intercept is…"; "The slope tells me… and the y-intercept tells me…".
Close (5 minutes) — ACE
- Articulate: explain the difference between evaluating f(x) at a value and solving f(x) = k. Which gives an output, and which gives an input?
- Connect: name one packet item where you used function notation f(x) to find an output.
- Extend: give a new real-life non-proportional linear function (a starting amount plus a steady rate) not used in this packet.
Continue (optional) — Early finisher
Write a context for the function f(x) = 7x + 20: invent a real-world situation the function could describe (say what x and f(x) stand for), explain what the slope 7 and the y-intercept 20 mean in your story, evaluate f(5), and solve f(x) = 90. Make sure your story is non-proportional (it must have a starting amount).
Turn in
Hand in the whole packet with your name, class period, and date, with items 1–6, 8, and the ACE box answered and your work shown (item 7 too if you are on a block schedule). Include the optional challenge if you did it.