Building Linear Function Models

Lesson 1 of 2: Constructing y = mx + b

In this lesson:

• Find the rate of change and initial value from any source
• Build the equation from words, tables, graphs, and points

What You Will Be Able to Do

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

1. Determine the rate of change from a description, table, graph, or two points
2. Determine the initial value, including working backward when is missing
3. Construct the equation from any representation

A Question About a Dog-Walking Bill

A neighborhood dog-walking service charges:

• a $5 flat fee every week, plus
• $2 for each walk

If you booked any number of walks, could you predict the bill?

From Counting to a Rule

A fixed $5, plus $2 once per walk.

Rate of Change: A Per-Unit Ratio

The rate of change is how much the output changes per one-unit change in input.

• It's always a ratio:
• "$2 per walk" means

Even from words, hear it as a "per unit" rate.

Initial Value and the Two-Step Build

The initial value is the output when the input is zero — the starting point.

To build any linear model:

1. Find the rate of change
2. Find the initial value
3. Write

Build It From a Taxi Fare

"A taxi costs $3.50 when you get in, plus $1.25 per mile."

• Rate : cost per mile
• Start : cost at zero miles
• Equation:

Check: at , .

Build It From a Water-Tank Table

The tank's water (gallons) over time (hours):

Row at gives directly; per hour.

When the Table Skips x = 0

A gym membership's total cost by month — no row:

; then .

Quick Check: Where Is b Hiding?

Look at a data table before you do anything.

• Does it include a row where ?
• If yes → read straight off that row
• If no → find first, then compute

Your Turn: Two Quick Builds

A. "A plumber charges $60 to show up, plus $40 per hour." Write .

B. Table: . Write .

Find , then , then the equation. Advance for answers.

What If b Isn't One of Your Points?

So far came from a table row or a fee in the story.

But a graph or a pair of points may never hand you .

Then you find first — and substitute a known point to solve for .

Build It By Reading a Graph

Summer-job earnings vs. hours worked:

; at the axis.

Example: Two Points, No Intercept

A line passes through and . Neither is the intercept.

• Rate:
• Start: use :
• Equation: (check: )

Quick Check: Why Must You Substitute?

You found from and .

Why can't you just read off these two points the way you did from a table row?

Build It From a Stretching Spring

"A spring is 14 cm with no weight. A 3 kg weight stretches it to 20 cm."

Points : , , so .

Three Common Traps to Avoid

• Swapped roles: "$30/month plus $50 fee" is , not
• First y-value as b: only valid if is in the table
• Forgetting to divide: slope is a ratio — , not

Quick Check: Find b Alone

A line has rate of change and passes through .

On your own: find . Write it down before advancing.

Your Turn: Build It All

A table of total savings, no row:

Find (it's negative!), find , write . No hints.

Key Takeaways From This Lesson

✓ Every model is the same 3 steps: find , find , write
✓ No given? Compute — never read the first value
✓ Slope is a ratio: divide by the change in

Decide vs. by role (per-unit vs. fixed), not sentence order

What Is Coming Up Next Lesson

You can now build the equation from any source.

But that's only half the standard — next lesson asks what and actually mean in the situation, and how to predict with your model.