Find the rate and start from (1, 7), (3, 15), (5, 23)
- Each x-step is 2 and each y-step is 8. Rate = 8 ÷ 2 = 4.
- Use y = 4x + b with (1, 7): 7 = 4 + b.
- b = 3, so y = 4x + 3.
Rate of change & starting value
Read what changes and what was already there at the start.
Lesson 07 · Start where you need
Your learning goal
Calculate change in output per unit input and distinguish the rate from the starting value.
Compare changes in both quantities
Rate of change is change in output divided by change in input. If a table moves from (1, 7) to (3, 15), y rises by 8 while x rises by 2. The rate is 8 ÷ 2 = 4 output units per input unit. A difference in y alone is insufficient when the x-steps are not 1.
Words worth knowing
- Ordered pair
- An input and its output, written (x, y).
- Rate of change
- Change in output divided by change in input.
- Starting value
- The output when the input is zero.
- Domain
- The set of permitted inputs for the situation.
1
Make the idea visible.
Predict a change, adjust one value, then explain what stayed the same.
The line represents a rule. Use the context to decide which inputs are meaningful.
y = 3x + 8
The same rule in a table
| x | y |
|---|
Find the value at zero input
In y = mx + b, m is the rate and b is the output at x = 0. The first row of a table may have x = 1 or x = 5; its output is not automatically b. Use the rule to step back to zero, or substitute a known pair to find the starting value.
Describe the sign in context
A positive rate means y increases as x increases. A negative rate means y decreases. A zero rate gives a constant output. The units matter: a change of −2 litres per minute describes a tank losing water. It does not mean the amount in the tank is already negative.
Use a straight-line model within its limits
A tank starting with 20 L and draining at 2 L per minute follows V = 20 − 2t until it empties at 10 minutes. Extending the equation to 12 minutes gives −4 L, which is outside this physical model’s useful domain. The equation can be mathematically evaluated there without describing a possible tank volume.
Khan Academy
Watch with a purpose
Linear equations in slope-intercept form
Identify the rate and the starting value before the explanation. Check whether the example’s input and output have units.
Watch on YouTube →Use captions when available. The lesson’s explanations and worked examples cover the idea in text.Worked examples
Follow the reasoning.
Interpret y = −2x + 20 as a tank model
- Starting volume: 20 litres at 0 minutes.
- Rate: −2 litres per minute.
- At 10 minutes the model gives 0 litres. Restrict the physical domain to 0 ≤ x ≤ 10.
Try it with support
Show your thinking.
Find the rate from (2, 9) to (5, 18).
Give me a hint
Divide the change in y by the change in x.
Compare a worked response
3
The output change is 9 and the input change is 3: 9 ÷ 3 = 3.
A table begins at (4, 17) and has rate 3. Find b.
Give me a hint
Use 17 = 3(4) + b.
Compare a worked response
b = 5
The value at x = 4 includes four rate increments. Subtract 12 to get the starting value.
Independent practice
Check your understanding.
Fresh questions · useful feedback
Build fluency, one decision at a time.
Work through eight questions. Two independent correct answers introduce a harder version; supported answers bring the next question back to the foundation. You can use hints at any time.
0 of 8 questions completed
This round saves on this device. It is practice, not a school mark.
Check the concepts, too.
Choose a response, then use the explanation to check your reasoning. This is practice, not a school mark.
Go further
Keep the connections close.
Khan Academy: graphing lines & slope → · All related learning collections →