Lesson 18: Linear Relationships: Tables, Graphs, and Slope Preview

A plant is 4 cm tall when you buy it, and it grows 3 cm every week. Week 2: 10 cm. Week 4: 16 cm. This steady, predictable growth is a linear relationship — the bridge between the proportional relationships of the last lesson and the algebra of lines you will meet next year. In this lesson you will read linear patterns in tables, compute slope as a rate of change, find the starting value (the y-intercept, in preview), and use both to make predictions.

Patterns in tables

Look at the plant table: x (weeks): 0, 1, 2, 3; y (height): 4, 7, 10, 13. Each week adds exactly 3 — a constant difference, the signature of a linear relationship. A phone plan shows the same shape: data used 0, 1, 2, 3 GB costs $20, $25, $30, $35 — each gigabyte adds $5. Constant differences in a table mean a straight-line graph, always.

Slope: the rate of change

The slope measures how fast y changes as x increases by 1: slope = (change in y) ÷ (change in x), also called "rise over run".

Worked example 1. Between the points (1, 2) and (3, 8): change in y = 8 − 2 = 6; change in x = 3 − 1 = 2; slope = 6 ÷ 2 = 3.

Worked example 2. Between (0, 1) and (4, 9): change in y = 8, change in x = 4, slope = 8 ÷ 4 = 2. Notice both examples have positive slope — the lines climb as you read left to right.

Worked example 3 (negative slope). A water tank drains: at time 0 it holds 40 gallons; after 2 hours, 34; after 4 hours, 28. Slope = (28 − 40) ÷ (4 − 0) = −12 ÷ 4 = −3 gallons per hour. The negative sign reports decrease — the line falls. The tank empties in 40 ÷ 3 ≈ 13.3 hours at this rate.

The starting value and the full pattern

Every linear relationship here has the form y = (starting value) + (slope) × x. The starting value is y when x = 0 — where the graph crosses the y-axis (the y-intercept, in preview language). The plant: y = 4 + 3x. The phone plan: cost = 20 + 5 × (GB). Lesson 17's proportional relationships are the special case where the starting value is 0 — one family, two members.

Worked example 4 (prediction). Plant height after 6 weeks: y = 4 + 3 × 6 = 4 + 18 = 22 cm. Prediction is the whole point of modelling: measure the pattern, write the rule, jump ahead.

In the real world: taxis, tanks and temperatures

Taxi fares (flag-drop fee plus per-mile rate), a kettle cooling down (negative slope), a candle burning (height decreasing steadily), and a savings account with regular deposits all trace straight lines. Engineers sketch these graphs before building anything: the slope tells them the rate they must design for, and the starting value tells them the initial condition. When a graph in a news story is a straight line, someone is claiming a constant rate — now you can check the claim against the numbers.

Common mistakes

• Dividing the changes the wrong way round. Slope is change in y over change in x. Between (2, 3) and (6, 11): (11 − 3) ÷ (6 − 2) = 8 ÷ 4 = 2, not 4 ÷ 8.
• Mismatching the pairs. Subtract the y-values in the same order as the x-values. If you compute 3 − 11 for y, you must compute 2 − 6 for x — the slope comes out the same (+2) only when the orders match.
• Ignoring the starting value. A table starting at y = 5 when x = 0 with slope 3 gives y = 5 + 3x, not y = 3x. Predicting with the wrong rule at x = 2 gives 6 instead of 11.
• Reading slope as "the y-value". Slope is a rate with units (cm per week, dollars per GB). A slope of 5 on the phone plan means $5 per gigabyte, not $5 total.

Exam Experience

1. What is the slope of the line through (2, 3) and (6, 11)? A) 1/2 B) 2 C) 4 D) 8

2. The relationship y = 3x − 4 crosses the y-axis at which value? A) 3 B) −3 C) 4 D) −4

3. Which table shows a linear relationship with slope 2 and starting value 1? A) x: 0, 1, 2 → y: 1, 2, 3 B) x: 0, 1, 2 → y: 1, 3, 5 C) x: 0, 1, 2 → y: 2, 4, 6 D) x: 0, 1, 2 → y: 1, 4, 9

4. (Constructed response) A plant is 4 cm tall when measured at week 0. At week 2 it is 10 cm tall, and at week 4 it is 16 cm tall. It grows at a constant rate. (a) Calculate the slope, in cm per week, and explain what it means. (b) Write the rule connecting height h and weeks w. (c) Predict the height at week 6. (d) The plant must reach 40 cm before it can be planted outside. In which week will it first reach at least 40 cm?

Answer key and explanations

• 1. B — (11 − 3) ÷ (6 − 2) = 8 ÷ 4 = 2.
• 2. D — The starting value (y when x = 0) is the constant term, −4.
• 3. B — Starting value 1, and each step adds 2: 1, 3, 5.
• 4. (a) Slope = (10 − 4) ÷ (2 − 0) = 6 ÷ 2 = 3 cm per week — the plant grows 3 cm each week. (b) h = 4 + 3w. (c) h = 4 + 3 × 6 = 22 cm. (d) 4 + 3w ≥ 40 → 3w ≥ 36 → w ≥ 12, so it first reaches 40 cm at week 12 (h = 4 + 36 = 40 exactly).

Scoring guide for question 4 (4 points)

• 1 point for the slope, 3 cm per week, with its meaning stated (growth per week).
• 1 point for the rule h = 4 + 3w, using the starting value 4.
• 1 point for the prediction, 22 cm.
• 1 point for week 12, reached by solving or trialling the rule (4 + 3 × 12 = 40).

Going further: slopes from tables and tricky points

Worked example 5. A table: x: 0, 2, 4, 6; y: 3, 7, 11, 15. The y-values climb by 4 while x climbs by 2, so the slope is 4 ÷ 2 = 2 per unit of x. The rule is y = 3 + 2x, and at x = 10 the prediction is 3 + 20 = 23. When table steps are not 1, divide the y-change by the x-change — never just quote the y-jump.

Worked example 6 (negative coordinates). Find the slope between (−1, 4) and (2, −2). Change in y: −2 − 4 = −6. Change in x: 2 − (−1) = 3. Slope = −6 ÷ 3 = −2. Both changes must be taken in the same point order, and subtracting a negative in the x-change is where most errors hide — write the subtractions out in full, exactly as here.

Worked example 7 (flat line). Between (1, 5) and (4, 5): change in y = 0, so the slope is 0. A horizontal line means y is not changing at all: a tank neither filling nor draining, a car parked, a price frozen. Slope 0 is information, not an absence of slope. (The opposite — a vertical line, where x never changes — has no defined slope, because the x-change is 0 and division by 0 is impossible.)

Worked example 8. A gym charges a $30 joining fee plus $8 per class. Cost rule: C = 30 + 8s. The slope, 8, is the price per class; ten classes cost 30 + 80 = $110. Slope in a money context is always a unit price — which ties this lesson back to lesson 6's unit rates. One idea, many costumes.

Quick practice

• 1. Slope between (0, 7) and (3, 1).
• 2. The y-intercept (starting value) of y = 5x + 9.
• 3. Evaluate y = 2 + 4x at x = 5.
• 4. A table has y-values 10, 14, 18 as x increases by 1 each time. Is the relationship linear, and what is the slope?

Answers. 1. (1 − 7) ÷ 3 = −2. 2. 9 (the value when x = 0). 3. 2 + 20 = 22. 4. Yes, linear — constant difference 4, so slope = 4.

Investigation: the temperature line

Weather apps quietly convert between Celsius and Fahrenheit all day. Plot the known pairs: (0°C, 32°F), (10, 50), (20, 68), (100, 212). The slope between the first two: (50 − 32) ÷ (10 − 0) = 18 ÷ 10 = 1.8 °F per °C. The starting value is 32, so the rule is F = 32 + 1.8C. Test it on the last pair: 32 + 1.8 × 100 = 32 + 180 = 212 ✓ — boiling point, confirmed. Now the famous one: human body temperature, 37°C, becomes 32 + 1.8 × 37 = 32 + 66.6 = 98.6°F — the number American thermometers are built around. And a slope observation worth keeping: 1.8 is not an integer, so Fahrenheit steps do not land on round Celsius marks; conversion tables zigzag forever. This investigation is the whole linear toolkit earning its keep — read two points, compute the rate of change, anchor the start value, write the rule, verify it against a third point, then use it somewhere the answer matters.

Challenge corner

Find the slope between (−2, 5) and (4, −1), a line falling through negative territory: change in y is −1 − 5 = −6; change in x is 4 − (−2) = 6; slope = −6 ÷ 6 = −1. A slope of exactly −1 means y drops by precisely the amount x rises — the line is a perfect downhill diagonal. If this line passes through (0, 3), its rule is y = 3 − x (starting value 3, slope −1). Predict y at x = 10: 3 − 10 = −7. Negative predictions are meaningful — a temperature, a bank balance, a depth — and this course has been preparing you for them since lesson 1's integers. Slope, intercepts and rules are about to become the full study of linear equations in Algebra 1; you are arriving with the machinery already assembled and the intuition already warmed up.