The ratio of the increase in y to the increase in x is called the rate of change. For a linear function y=ax+b, the rate of change is always equal to a（slope）.
Formula for the Rate of Change
Rate of change ＝ increase in y ÷ increase in x ＝ Δy / Δx
Example ①
For y = 2x + 3, find the rate of change when x increases from 1 to 4.
Solution:x=1→y=5、x=4→y=11. Rate of change = (11−5)/(4−1) = 6/3 = 2（= slope a）
For y = 2x + 3, find the rate of change when x increases from 1 to 4.
Solution:x=1→y=5、x=4→y=11. Rate of change = (11−5)/(4−1) = 6/3 = 2（= slope a）
Example ②
Find the equation of the linear function whose rate of change is −3 and whose y-value is 1 when x=2.
Solution:Since a=−3, y=−3x+b. Substitute x=2,y=1: 1=−6+b → b=7。y=−3x+7
Find the equation of the linear function whose rate of change is −3 and whose y-value is 1 when x=2.
Solution:Since a=−3, y=−3x+b. Substitute x=2,y=1: 1=−6+b → b=7。y=−3x+7
Key Points
- Rate of change of a linear function = a（always constant）
- Knowing the rate of change determines a
- Knowing a and one point determines b
Practice Problems
- For y = −2x + 5, find the rate of change when x increases from −1 to 3.
- Find the equation of the linear function whose rate of change is 4 and whose y-value is −3 when x=0.
Answers and Explanations
- Answer:−2（equal to the slope）.
- Answer:y=4x−3. a=4,b=−3（because y=b when x=0）.