Construct a linear function to model a relationship · lesson

Start value plus rate

💡 Amount = start value + rate × time, with a negative rate for things that go down.

Many situations start at some amount and change by the same amount each step.

Amount = start + rate × steps. If it goes down, the rate is negative.

🧠 Let's work it out

A tank has 30 gallons and fills at 12 gallons per minute. How much after 5 minutes?

What do you think comes next?

💡 The rate goes with the variable and the one-time amount is the constant.

A one-time fee or a starting amount is the constant. A per-hour or per-month amount is the rate, multiplied by the variable.

If the amount goes down, subtract the rate times the variable.

🧠 Let's work it out

A tutor charges a $35 fee plus $25 per hour. Write the cost C of h hours.

What do you think comes next?

💡 The rate of change is the change in y divided by the change in x.

If you know two points, the rate is how much y changed divided by how much x changed.

This is the same as the slope.

🧠 Let's work it out

A line goes through (1, 12) and (4, 27). What is the rate of change?

What do you think comes next?

💡 Use the rate and one point to work back to the value at x = 0.

The initial value is y when x = 0. It is the b in y = mx + b.

Find the rate first. Then step back from a known point to x = 0, or solve y = mx + b for b.

🧠 Let's work it out

A line goes through (1, 12) and (4, 27), with rate 5. Find its equation.

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💡 Set the model equal to the target amount and solve for the variable.

To ask when a model reaches some amount, set the equation equal to that amount.

"Empty" means the amount is 0.

🧠 Let's work it out

A cab costs $4 plus $3 per mile. How far can you go for $25?

Write the equation: 4 + 3m = 25.
Subtract 4: 3m = 21.
Divide by 3: m = 7 miles.
Another example: a 180-gallon tank draining 12 gallons a minute is empty when 180 − 12t = 0, so t = 15.

What do you think comes next?