Equation, Slope, and Intercept

You will be able to:

• Write for a fitted line
• Interpret slope and intercept in context
• Make and classify predictions

Learning Objectives for This Lesson

By the end of this lesson, you will be able to:

1. Write the equation for an informally fitted line
2. Interpret the slope in context as a rate with units
3. Interpret the y-intercept and judge whether is meaningful
4. Make predictions and classify them as interpolation or extrapolation

What Does the 1.5 Actually Mean?

A biology class fit . What does a scientist learn from "1.5"?

Start With Slope: Rise Over Run

• Pick two points on the line: and

Find the Intercept, Write the Equation

Step 1: Use a point on the line:

Step 2: Solve: , so

Step 3: Equation:

The Equation Describes the Line

• The equation matches the line, not the dots
• Real data points scatter above and below it
• Different fitted lines give slightly different equations

Slope Is a Rate, With Units

• Slope means cm per hour of sunlight
• Units are always "[units of ] per [unit of ]"
• It tells how fast changes as changes

The Slope Sentence Frame to Use

Use this structure every time:

"For each additional [unit of ], [] is associated with a change of [] [units]."

For our plants: each extra hour of sunlight → about 1.5 cm more height.

Quick Check: A Two-Hour Increase

Sunlight increases by 2 hours. Using slope 1.5, what height change does the model predict?

Work it out before advancing.

Say "Associated With," Not "Causes"

• The model shows sunlight and height move together
• That is an association, not proven cause
• Use careful language: "associated with"

A Negative Slope Tells the Opposite Story

• Car age vs. resale:
• Each extra year → about $1,800 less in value

From Slope to the Y-Intercept

The intercept is the line's value at .

But that only means something if makes sense for the data.

When an Intercept Is Meaningful

• Plant model:
• At hours of sunlight, predicted height is cm
• Maybe the seedling's starting height — but is outside our data

Three Intercepts and Three Verdicts

• Meaningful: study hours = 0 → a real score
• Possible: plant at 0 sunlight → maybe seedling height
• Absurd: age 0 → newborn earning $8,000/year

Two Questions for Any Intercept

Always ask:

1. What does mean in this context?
2. Is near the data, or far outside it?

Interpolation: Predicting Inside the Data

Plant model: , data from 3 to 10 hours

Predict at : cm

6.5 is inside the range — this prediction is trustworthy.

Extrapolation: Predicting Beyond the Data

Predict at hours: cm

• 15 is beyond our 3-to-10 range
• The trend might not continue — treat with caution

When the Model Clearly Breaks Down

• At 20 hours: cm
• But a day has only 24 hours, and plants need darkness
• Far extrapolation produces unreasonable results

Quick Check: Classify a Prediction

Car model: , data from 3 to 15 years.

Predict resale of a 5-year-old car. Is it interpolation or extrapolation?

Decide, then advance.

Comparing Positive and Negative Slope

Compare two models with the sentence frame:

• Plants: each hour → height increases 1.5 cm
• Cars: each year → price decreases $1,800

What single word in the sentence flips with the sign?

Sort These Intercepts by Meaning

Which of these values are meaningful?

• Hours studied
• Building height stories → number of elevators
• Age → annual salary (data: ages 25–65)

Your Turn: Interpret a New Model

A model for typing speed: , where is weeks of practice.

1. Interpret the slope in context
2. Interpret the intercept — is meaningful?
3. Predict at and classify it

Do all three on your own first.

Three Interpretation Traps to Avoid

Slope is a y-value: slope is a change per unit, not how tall is
Intercept always means something: ask if makes sense first
Predictions are exact: they are estimates from a fitted line

Numbers Become Claims About the World

✓ Slope is a rate with units, stated "for each additional..."
✓ Intercept is at — meaningful only if fits
✓ Predictions are trustworthy inside the data, risky outside

Coming Up Next: A Full Analysis

Next lesson, you'll run the whole analysis on one fresh data set — equation, slope, intercept, predictions — and confront whether the data support a causal story.