Linear Functions in Four Forms
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This is a paper math packet about linear functions — functions whose graphs are straight lines. You will connect the same function shown four ways: a table, a graph, an equation in function notation f(x), and a real-world context. You will also write equations from information and solve equations such as f(x) = a. Work by yourself with a pencil. A calculator is not required, but a basic or graphing calculator is allowed if you have one. Show your work — every answer should show the numbers and steps you used, not just a final number. Everything you need (formulas, a worked example, a graph) is printed right here. If a question is hard, skip it, keep going, and come back. Circle or box your final answers.
Start Warm-Up: Retrieval 6 min
Bring back what you already know about evaluating a function and finding slope.
1Evaluate a function. A function is defined by f(x) = 2x + 3. Find f(4). Show the multiplication and the addition you used. (Remember: f(4) means "the output when the input x is 4" — it does not mean f times 4.)
2Find slope from two points. A line passes through the points (1, 5) and (4, 11). Find the slope m. Show m = (y₂ − y₁) ÷ (x₂ − x₁) with your numbers.
Build Study the Math 8–12 min
What you need to know (with formulas)
A linear function has a graph that is a straight line. Its output changes by the same amount for each equal step of the input.
Function notation. We write f(x) ("f of x") for the output of the function at input x. So f(4) is the output when x = 4. The notation f(x) is not multiplication — it names the output, not "f times x."
The slope (written m) is the rate of change: how much the output changes for each 1 that x increases. From two points (x₁, y₁) and (x₂, y₂):
m = rise ÷ run = (y₂ − y₁) ÷ (x₂ − x₁)
The y-intercept (written b) is the output when x = 0 — the point (0, b) where the line crosses the vertical axis. The x-intercept is where the line crosses the horizontal axis; it is the value of x that makes the output 0 (solve f(x) = 0). Slope and y-intercept give the slope-intercept form:
y = mx + b or, in function notation, f(x) = mx + b
These say the same thing: y and f(x) both name the output. Use "y" when you graph an ordered pair (x, y); use "f(x)" when you want to talk about the function's output at a named input.
A linear function can be shown four ways, and all four match the same m and b:
- Table: as x goes up by a fixed step, the output goes up (or down) by a fixed step. m = (change in output) ÷ (change in x). The row where x = 0 gives b.
- Graph: a straight line. It crosses the y-axis at (0, b) and rises m units for each 1 unit right.
- Equation: f(x) = mx + b.
- Context: a real situation with a starting amount (b) and a steady rate of change (m) per one unit.
Solving f(x) = k. To find the input x that gives a chosen output k, replace f(x) with k and solve the linear equation:
mx + b = k → mx = k − b → x = (k − b) ÷ m
Tip: "evaluate" means you are given x and you find the output. "Solve" means you are given the output and you find x. They are opposite directions.
Worked Example — one function, four forms
A tutoring service charges a flat booking fee plus a steady hourly rate. The cost starts at $3 (a booking fee) and rises $2 for every hour. Let x = hours and let f(x) = total cost in dollars. Because there is a starting amount of 3, this line does not pass through the origin.
Slope (rate of change): m = 2 dollars per hour. y-intercept: b = 3 dollars (the cost at 0 hours).
Equation (function notation): f(x) = 2x + 3.
Check the slope from two table rows: using (1, 5) and (3, 9), m = (9 − 5) ÷ (3 − 1) = 4 ÷ 2 = 2. ✔
Evaluate: f(4) = 2(4) + 3 = 8 + 3 = 11, so 4 hours costs $11.
Solve f(x) = 15: 2x + 3 = 15 → 2x = 12 → x = 6, so $15 buys 6 hours.
| Hours x | Cost f(x) (dollars) | Change in output |
|---|---|---|
| 0 | 3 | — (start) |
| 1 | 5 | +2 |
| 2 | 7 | +2 |
| 3 | 9 | +2 |
| 4 | 11 | +2 |
3Using the worked example, find f(7) (the cost at 7 hours). Use f(x) = 2x + 3 and show your work.
Apply Use What You Know 20–26 min
Show the numbers you use in every item. A calculator is allowed.
4Match the four forms. A phone-repair shop charges a diagnostic fee plus a fixed price for each part used. The same linear function is shown below as a table, a graph description, an equation, and a context. Study Table B, then answer 4a–4e.
| Parts x | Total cost g(x) (dollars) |
|---|---|
| 0 | 25 |
| 2 | 55 |
| 4 | 85 |
| 6 | 115 |
Context: "A repair shop charges a one-time $25 diagnostic fee, then $15 for each part used. Total cost = diagnostic fee + $15 times the number of parts."
4a. Find the slope m (dollars per part). Use two rows of the table and show m = (change in output) ÷ (change in x).
4b. Find the y-intercept b (the 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 the words "per part" and "diagnostic fee."
4d. Solve g(x) = 145. A repair cost a total of $145. How many parts were used? Set up your equation and solve for x, showing every step.
4e. Is this function proportional or not proportional? Explain using the y-intercept in one sentence.
5Write the equation from a context. A gym membership starts with a $40 sign-up fee and then costs $12 per month. Let x = number of months and let h(x) = total cost in dollars.
5a. Identify the slope m and the y-intercept b from the context, then write h(x) = mx + b.
5b. Use your equation to find h(9), the total cost after 9 months. Show your work.
6Error analysis (a mis-found slope). A student found the slope for Table B and wrote the work shown. Find the mistake, explain it, and give the correct slope.
Student's work (using the rows (2, 55) and (4, 85)):
m = run ÷ rise = (4 − 2) ÷ (85 − 55)
m = 2 ÷ 30 ≈ 0.07. So the slope is about 0.07 dollars per part.
What is wrong, and what is the correct slope? Fix it in the box.
Block Extra Applied Task block only · ~12–15 min
7Real-world modeling. A water tank begins with 60 gallons and is draining at 4 gallons per minute. Let x = minutes and let V(x) = gallons of water left in the tank.
7a. Write the function V(x) = mx + b. (Careful: the water is going down, so the rate of change is negative.)
7b. Find V(6) — how much water is 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 what the x-intercept means in this story.
Explain Justify Your Strategy 6–10 min
Question 8. Two streaming services charge a monthly bill. Plan A: f(x) = 3x + 8 (an $8 base charge plus $3 per premium channel). Plan B: g(x) = 5x (just $5 per premium 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 ACE Wrap-Up 5 min
In your own words, explain the difference between evaluating f(x) at a value and solving f(x) = k. Which one gives an output, and which gives an input?
Point to one item in this packet where you used function notation f(x) to find an output. Give its number.
Give a new real-life example of a non-proportional linear function (a starting amount plus a steady rate) that was not in this packet.
Continue Early Finisher optional
Write a context for a function. Here is a function with no story: f(x) = 7x + 20. On the back of this page: (1) invent a real-world situation that this function could describe, clearly telling what x and f(x) stand for; (2) explain what the slope 7 and the y-intercept 20 mean in your story; (3) evaluate f(5), showing your work; and (4) solve f(x) = 90, showing your steps. Make sure your story is non-proportional (it must have a starting amount).