Read the slope and intercept from a graph to find its equation. Also, find the range of y from the range of x.
Finding an Equation from a Graph
Example ①
Find the equation of the line passing through the points (0, 3) and (2, 7).
Answer: Intercept b=3 (when x=0, y=3). Slope a=(7−3)/(2−0)=2. y=2x+3
Find the equation of the line passing through the points (0, 3) and (2, 7).
Answer: Intercept b=3 (when x=0, y=3). Slope a=(7−3)/(2−0)=2. y=2x+3
How to Find the Range
Example ②
For y = 2x − 1, find the range of y when −1 ≦ x ≦ 3.
Answer: Since a=2 > 0, y also increases as x increases. x=−1: y=−3, x=3: y=5. −3 ≦ y ≦ 5
For y = 2x − 1, find the range of y when −1 ≦ x ≦ 3.
Answer: Since a=2 > 0, y also increases as x increases. x=−1: y=−3, x=3: y=5. −3 ≦ y ≦ 5
Key Points
- a > 0 → y is smallest when x is smallest, and y is largest when x is largest
- a < 0 → y is largest when x is smallest, and y is smallest when x is largest
- Substitute the endpoint values of the range to find the answer
Practice Problems
- For y = −3x + 4, find the range of y when 0 ≦ x ≦ 2.
Answers and Explanations
- Answer: −2 ≦ y ≦ 4. a=−3 < 0 → when x=0, y=4 (maximum), and when x=2, y=−2 (minimum).