Reading Rate and Starting Value

Lesson 2 of 3: Functions (Deck 1 of 2)

In this lesson:

• Find the rate of change in any form
• Find the initial value — even by working backward

Learning Objectives: Rate and Initial Value

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

1. Find the rate of change from an equation, table, graph, or verbal description
2. Find the initial value from any of those four representations
3. Work backward to find the initial value when is missing from a table

Who Is Saving Money Faster?

Aisha puts away $15 every week.

Ben already has some money and adds $10 every week.

Who is saving faster? You already know.

The Same Saver, Three Ways

Ben's balance, three ways:

• Words: starts at $40, adds $10 each week
• Equation:
• Table:

The "10 a week" is hiding in all three.

Rate of Change, Defined Precisely

The rate of change is how much the output changes per unit of input:

Always a fraction — even when the bottom is 1.

Find the Rate From an Equation

In , the rate of change is the number multiplying :

Each time goes up by 1, goes up by 4.

From a Table, Steps of 1

From a Table, Steps of 2

Find the Rate From a Graph

Find the Rate From Words

• "A pool fills at 50 gallons per minute" rate
• "The temperature falls 2° each hour" rate

Listen for "per" — and watch the direction.

Quick Check: Find Each Rate

2. Table:
3. "A car uses 2 gallons every 50 miles"

Find the rate of change for each. Watch the fractions.

But Where Did Ben Start?

We know Ben adds $10 a week. But how much did he have before week one?

That second number is the initial value.

The Initial Value, Defined Precisely

The initial value is the output when the input is zero.

In , it's the number .

It's the starting amount — the fixed fee, the head start, the baseline.

From an Equation or a Ready Table

Equation : initial value is .

Table with :

Initial value .

When Is Missing: Work Backward

Verify by Writing the Equation

Rate , initial value , so:

Check at : ✓

From a Graph or a Description

Graph: read where the line crosses the -axis (at ).

Description: "A plumber charges $75 plus $40 per hour" initial value .

Find the Error in This Reading

A student reads this table and says "the rate of change is 16":

What did they forget? What's the real rate?

Your Turn: Decide on Your Own

On your whiteboard, find the rate of change:

Write the full fraction. No calling out.

Find Both Numbers, Any Form

For each, give the rate of change and initial value:

2. Table:
3. Graph crossing , through
4. "Tank starts at 100 L, drains 5 L/min"

Three Common Traps to Avoid

Always divide — the -jump is the rate only when steps by 1

The first table row isn't the initial value unless

Label what you extract — rate and initial value are different numbers

Two Numbers in Any Packaging

✓ Every linear function carries two numbers: rate of change and initial value

✓ You can pull both from an equation, table, graph, or words

Next: put two functions side by side — which is bigger?