Linear Relationships in Four Forms
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This is a paper math packet about linear relationships — the kind that make a straight line. You will connect the same relationship shown four ways: a table, a graph, an equation, and a real-world story. Work by yourself with a pencil. A calculator is not required, but a basic calculator is allowed if you have one. Show your work — every answer should show the numbers 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 5 min
Bring back what you already know about rules and reading a graph.
1Evaluate a rule. A rule says y = 2x + 5. Find the value of y when x = 4. Show the multiplication and the addition you used.
2Read a point off a line. On the graph in the Build section (Figure 1), the line passes through several marked points. Find the point where x = 2 and write it as an ordered pair (x, y). (You may look ahead to Figure 1 to answer this.)
Build Study the Math 5–10 min
What you need to know (with formulas)
A linear relationship between two quantities x and y makes a straight line when graphed. It changes by the same amount each step.
The rate of change (also called the slope, written m) tells how much y 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 value of y when x = 0 — the point (0, b) where the line crosses the vertical axis. Together they give the slope-intercept form of a line:
y = mx + b (m = slope, b = y-intercept)
A linear relationship can be shown four ways, and all four match the same m and b:
- Table: as x goes up by a fixed step, y goes up (or down) by a fixed step. m = (change in y) ÷ (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: y = mx + b.
- Situation: a real story with a starting amount (b) and a steady rate (m) per one.
Proportional vs. non-proportional. A linear relationship is proportional only when b = 0 — the line passes through the origin (0, 0) and y = mx. If b is not 0 (there is a starting amount), the line is linear but NOT proportional; it crosses the y-axis above or below the origin.
Tip: not every straight line is proportional. Check the y-intercept — only a line through (0, 0) is a proportional relationship.
Worked Example — one relationship, four forms
A pool already holds some water, then a hose adds water at a steady rate. The depth starts at 4 inches and rises 3 inches every minute. Because there is a starting amount of 4, this is linear but NOT proportional.
Slope (rate of change): m = 3 inches per minute. y-intercept: b = 4 inches (the depth at 0 minutes).
Equation: y = 3x + 4, where x = minutes and y = depth in inches.
Check the slope from two table rows: using (1, 7) and (2, 10), m = (10 − 7) ÷ (2 − 1) = 3 ÷ 1 = 3. ✔
Table (each step of 1 in x adds 3 to y) and graph below. The line crosses the y-axis at (0, 4) — not the origin — so it is non-proportional.
| Time x (minutes) | Depth y (inches) | Change in y |
|---|---|---|
| 0 | 4 | — (start) |
| 1 | 7 | +3 |
| 2 | 10 | +3 |
| 3 | 13 | +3 |
| 4 | 16 | +3 |
3Using the worked example, how deep would the water be at 6 minutes? Use y = 3x + 4 and show your work.
Apply Use What You Know 20–25 min
Show the numbers you use in every item. A basic calculator is allowed.
4Match the four forms. A gym charges a one-time sign-up fee plus a fixed price for each visit. The same linear relationship is shown below as a table, a graph description, an equation, and a story. Study Table B, then answer 4a–4d.
| Visits x | Total cost y (dollars) |
|---|---|
| 0 | 20 |
| 2 | 30 |
| 4 | 40 |
| 6 | 50 |
Story: "A gym charges a one-time $20 sign-up fee, then $5 for each visit. Total cost = sign-up fee + $5 times the number of visits."
4a. Find the slope m (dollars per visit). Use two rows of the table and show m = (change in y) ÷ (change in x).
4b. Find the y-intercept b (the cost at 0 visits), then write the equation y = mx + b for this relationship.
4c. Interpret in context. In this story, what does the slope mean, and what does the y-intercept mean? Use the words "per visit" and "sign-up fee."
4d. Is this relationship proportional or not proportional? Explain using the y-intercept in one sentence.
5Error 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, 30) and (4, 40)):
m = run ÷ rise = (4 − 2) ÷ (40 − 30)
m = 2 ÷ 10 = 0.2. So the slope is 0.2 dollars per visit.
What is wrong, and what is the correct slope? Fix it in the box.
6Multi-step application (predict a value from the model). Use the gym relationship y = 5x + 20 from item 4.
6a. Predict the total cost of 12 visits. Show y = 5(12) + 20.
6b. A member paid a total of $75. How many visits did they make? Set up 5x + 20 = 75 and solve for x. Show your steps.
Explain Justify Your Strategy 5–10 min
Question 7. Two plans charge for renting a bike. Plan A: y = 4x + 10 (a $10 sign-up plus $4 per hour). Plan B: y = 6x (just $6 per hour, no sign-up). For a 3-hour rental, 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 hours, 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 how to find the slope from two rows of a table using rise over run.
Point to one item in this packet where you used y = mx + b. Give its number.
Give a new real-life example of a non-proportional linear relationship (a starting amount plus a steady rate) that was not in this packet.
Continue Early Finisher optional · ~15 min
Write a context for an equation. Here is an equation with no story: y = 8x + 15. On the back of this page: (1) invent a real-world situation that this equation could describe, clearly telling what x and y stand for; (2) explain what the slope 8 and the y-intercept 15 mean in your story; and (3) use your model to predict y when x = 5, showing your work. Make sure your story is non-proportional (it must have a starting amount).