Constructing Functions from Two Points and Tables

Lesson 1 of 2: Two-Point Methods and Tables

In this lesson:

• Construct linear functions from two input-output pairs
• Construct exponential functions from two input-output pairs
• Read tables to identify and build both function types

What You Will Learn Today

1. Construct a linear function from two points
2. Construct an exponential function from two points
3. Build functions from tables using difference and ratio tests
4. Connect arithmetic and geometric sequences to function forms

Two Points Pin Down One Line

Two points determine a unique line. The slope connects them; the y-intercept anchors it.

Finding Slope from Two Ordered Pairs

Points and :

Slope = 3: for every 1-unit increase in , increases by 3.

Find b, Write the Function, Verify

Slope . Use point :

Function:

Verify: ✓ and ✓

Applying the Two-Point Method: Candle Height

"A candle is 12 in. tall after 1 hour and 9 in. tall after 3 hours."

Points: and

Tables with Constant First Differences

All differences = +3 → linear. , .

Arithmetic Sequences Connect to Linear Functions

Arithmetic sequence: (common difference )

Sequence notation:

Function notation:

The common difference is the slope . The sequence IS the function on integer inputs.

Quick Check: Construct a Linear Function

Points: and

1. Compute the slope.
2. Use to find .
3. Write the function.
4. Verify both points.

Try before the next slide.

Exponential Two-Point Method: The Key Idea

For with two points and :

Dividing eliminates . Solve for , then back-substitute for .

Exponential Example: Points (1, 6) and (3, 54)

Back-substitute into first equation:

— Verify: ✓ and ✓

Shortcut for Consecutive Integer Input Tables

When values are consecutive integers, the ratio of consecutive values equals .

Table: , , ,

Ratios: , , — constant

,

Differences vs. Ratios in a Table

Differences not constant → not linear. Ratios constant → exponential.

Context Example: Bacteria Colony Growth

"A colony has 200 cells at and 800 cells at hours."

Verify: ✓ and ✓

Geometric Sequences Connect to Exponential Functions

Geometric sequence: (common ratio )

Sequence:

Function: , or equivalently

Common ratio is the base . Sequence = function on integer domain.

Quick Check: Construct an Exponential Function

Points: and

1. Divide:
2. Solve: , so
3. Find using
4. Write and verify.

Try before the next slide.

Practice: Two Linear and Two Exponential

Linear:

1. Through and
2. Through and

Exponential:
3. Through and
4. Through and

For each: find and (or and ), write the function, verify.

Practice Answers: Four Construction Problems

1. , →
2. , →
3. , , →
4. , , →

Key Takeaways from Lesson One

✓ Linear: ; substitute for ; verify both points
✓ Exponential: divide outputs → ; solve for ; substitute for
✓ Table: constant differences → linear; constant ratios → exponential

Exponential: DIVIDE outputs — do NOT subtract to find the base
first table value if first input — solve for
Arithmetic sequence: . Geometric sequence: .

Coming Up in Lesson Two

Deck 2 covers:

• Reading graphs to extract two integer coordinate points
• Constructing from graphs (linear and exponential)
• Translating verbal descriptions into linear and exponential functions
• Mixed practice: classify and construct from any representation