How to Find X and Y Intercepts in Vertex Form
The vertex form of a quadratic equation, expressed as f(x) = a(x - h)² + k, is a powerful tool for analyzing parabolas. Worth adding: while it directly reveals the vertex (h, k), determining the x-intercepts (where the graph crosses the x-axis) and y-intercepts (where it crosses the y-axis) requires specific steps. This guide explains how to calculate both intercepts efficiently using the vertex form, with clear examples and common pitfalls to avoid.
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Understanding Intercepts in Vertex Form
What Are Intercepts?
- Y-intercept: The point where the parabola crosses the y-axis. At this point, x = 0.
- X-intercepts: The points where the parabola crosses the x-axis. At these points, f(x) = 0.
Vertex Form Basics
The vertex form f(x) = a(x - h)² + k encodes key features:
- (h, k): The vertex of the parabola.
- a: Determines the parabola’s width and direction (upward if a > 0, downward if a < 0).
Finding the Y-Intercept
To find the y-intercept, substitute x = 0 into the vertex form:
f(0) = a(0 - h)² + k = ah² + k
This gives the point (0, ah² + k).
Example:
For f(x) = 2(x - 3)² - 8:
- Y-intercept: f(0) = 2(0 - 3)² - 8 = 2(9) - 8 = 10
- Result: (0, 10)
