Fitting a Function to Data

Lesson 1 of 1: Choosing, Fitting, Using a Model

In this lesson:

• Choose a model family from a plot's shape
• Fit a function and use it to predict

Learning Objectives for This Lesson

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

1. Choose a model family from form and context
2. Fit a function — given or by regression
3. Use a fitted function to predict and solve
4. Distinguish interpolation from extrapolation

Turning a Table of Dots Into a Rule

A biologist records bacteria count vs hours — just dots on a plot.

• The table alone can't predict tomorrow's count
• Fit a function and the dots become a usable rule

But what kind of function? That choice comes first.

From Linear-or-Not to Three Families

You already read a plot's form as linear or nonlinear.

• "Nonlinear" is too vague to fit a function
• Refine it into two families: quadratic and exponential

Three shapes now: straight, turning, constant-percent.

The Linear Family: A Constant Rate

Phone-plan cost rises a fixed amount per minute — a straight line.

The Quadratic Family: A Turning Point

A thrown ball's height vs time rises, then falls.

• Plotted, it's a parabola — one turning point
• The mechanism is gravity, constantly bending the path

Rises then falls, with a single turning point → quadratic.

The Exponential Family: Constant Percent

Bacteria multiply each hour — count grows by a constant percent.

• The plot curves upward, steepening, never turning
• The mechanism is repeated multiplication

Constant-percent growth, a steepening curve → exponential.

The Three Model Family Signatures Compared

Same change · turning point · constant percent.

Your Turn: Match the Family

Name the family and justify from shape and mechanism:

1. Taxi fare: base fare + fixed cost per mile, straight band
2. Savings: fixed percent interest yearly, steepening curve

Name the family and say why, then advance.

Answer: 1. Linear (constant rate). 2. Exponential (constant percent).

From Family Name to an Equation

You named the family — but you still need an equation to predict.

• A name like "linear" isn't yet a usable rule
• Fitting produces the equation, by one of two routes

Next: the two routes to an equation.

Route 1: The Context Gives the Function

Sometimes the situation states the rule directly.

• A savings account growing 3% yearly → an exponential function
• A known per-minute rate → a linear function

When the mechanism is stated precisely, write the function from it.

Route 2: Technology Fits by Regression

No given formula? Regression finds the equation from data.

• LinReg, QuadReg, ExpReg on a calculator or spreadsheet
• You choose the family; technology returns the equation

The standard expects technology — no by-hand formula.

Regression on the Phone-Plan Data

Step 1: Enter the minutes-and-cost data, recognized as linear

Step 2: Run LinReg on the calculator

Step 3: Read the returned equation

Cost in dollars, x in minutes. We reuse this all lesson.

The Fit Captures the Trend, Not Every Point

• Points sit above and below — small gaps are normal
• Those gaps are what residuals (next lesson) measure

Your Turn: Fit a Function

Pick the route and produce the equation:

1. Tree: starts 2 ft, grows 1.5 ft/year — write it
2. Data table, straight band, no formula — how to fit?

State the route, then advance.

Answer: 1. (given). 2. LinReg → linear equation.

An Equation Is a Tool, Not the Goal

The equation exists to answer questions about the situation.

• Give it an x, predict a y
• Give it a y, solve for the x

Next: both directions — and how far you can trust them.

Predict y From x: Cost of 300 Minutes

Using , find the cost for 300 minutes.

Substitute :

The model predicts a $50 bill for 300 minutes.

Solve x From y: Minutes for $35

Using , find the minutes for a $35 bill.

Set and solve:

A $35 bill means 150 minutes used.

Interpolation Versus Extrapolation in Predictions

Inside the data range = interpolation (safe); outside = extrapolation (risky).

A Failing Extrapolation: The 30-Foot Adult

A line fit to a child's height vs age (ages 2–12) fits well.

• Extrapolate to age 100 → the line predicts a 30-foot adult
• The pattern only held inside the data range

The math gives a number; far outside the data, it's meaningless.

Quick Check: Interpolation or Extrapolation?

Phone-plan data covered about 0 to 250 minutes.

1. Predict the cost at 200 minutes
2. Predict the cost at 5,000 minutes

Classify each; which would you trust?

Answer: 1. Interpolation (in range) — trustworthy. 2. Extrapolation (far outside) — unreliable.

Find the Error: A Line on Curved Data

A student fit a line to data that clearly rises then falls.

• The line's predictions are off in a patterned way
• What family fits, and why?

Diagnose and fix it, then advance.

Answer: Quadratic — the data turns once; a line can't follow a turn. Refit with QuadReg.

Full Task: Choose, Fit, Predict, Flag

A city's population over 10 years curves steeply upward.

1. Name the family and justify it
2. State how you'd fit it
3. Make an in-range prediction; flag an extrapolation

Do all three, then advance.

Answer: Exponential (constant percent); ExpReg; predict within 10 years; 100 years out is unreliable.

Key Takeaways and What's Next

✓ Read shape + mechanism to choose the family

✓ Fit by a given function or by regression

✓ A fit captures the trend, not every point

Trust interpolation; scrutinize extrapolation

A model is an approximation, not a law

Next: assess the fit by analyzing residuals (6.b).