Relationship Between the Graph and the Axes
The relationship between the graph of a quadratic function and the x-axis is determined by the discriminant D, while the y-intercept is determined by the constant term c.
Relationship with the x-Axis
- D > 0 → intersects the x-axis at two points（has two x-intercepts）
- D = 0 → touches the x-axis（the vertex lies on the x-axis）
- D < 0 → does not intersect the x-axis
Example ①
Find the x-intercepts and y-intercept of y = x² - 3x + 2.
Answer: x-intercepts: x² - 3x + 2 = 0 → (x-1)(x-2) = 0 → x = 1, 2. y-intercept: x = 0 → y = 2.
Find the x-intercepts and y-intercept of y = x² - 3x + 2.
Answer: x-intercepts: x² - 3x + 2 = 0 → (x-1)(x-2) = 0 → x = 1, 2. y-intercept: x = 0 → y = 2.
Key Point
- x-intercepts: set y = 0 and solve
- y-intercept: set x = 0, giving y = c（the constant term）
Practice Problems
- Find the x-intercepts and y-intercept of y = x² - 5x + 4.
Answers and Explanations
- Answer: x-intercepts x = 1, 4. y-intercept y = 4.