Temperature vs Coffee

Year 9 · Linear Relationships

# Temperature vs Coffee

Finding the rule hidden in real data

---

## Lydia's coffee shop

The hotter the day, the **fewer coffees** Lydia sells. She records **5 days**:

<table class="dtable">
<tr><th>Max temp (°C)</th><td>15</td><td>18</td><td>27</td><td>31</td><td>39</td></tr>
<tr><th>Coffee sales</th><td>206</td><td>192</td><td>165</td><td>136</td><td>110</td></tr>
</table>

Can she turn this into a rule that predicts sales?

---

## Spot the trend

As the temperature **rises**, the sales **fall** — and steadily.

The points lie close to a straight line — a negative linear relationship. So let's find that line.

---

## Find the rule — the gradient

Use two points: (15, 206) and (39, 110).

sales drop by 4 for every extra °C — that matches the trend

---

## Find the rule — the intercept

Put a known point into

---

## Does the line fit?

Lydia's 5 days, with the rule C = −4T + 266 drawn through them

---

## Predict the sales

Hot day coming? Put the **temperature into** the rule.

---

## Work it backwards

Know the **sales**, want the **temperature**? Solve for *T*.

---

## When does it hit zero?

No coffee at all means **C = 0**:

…a 66.5 °C day? 🤔

---

## Can we trust that?

never happens here — and real sales never hit exactly **zero**.

The rule was built from days between 15 °C and 39 °C.

✅ Trust it inside that range — interpolation. 👈

❌ Don't stretch it far outside — extrapolation gives silly answers.