Solution: The vertex of a parabola $ h(x) = x^2 - 4x + c $ occurs at $ x = \frac{-(-4)}{2(1)} = 2 $, which matches the given condition. Now substitute $ x = 2 $ and set $ h(2) = 3 $:
Finding the Vertex of the Parabola $ h(x) = x^2 - 4x + c $: A Step-by-Step Solution
When analyzing quadratic functions, identifying the vertex is essential for understanding the graph’s shape and behavior. In this article, we explore how to find the vertex of the parabola defined by $ h(x) = x^2 - 4x + c $, using calculus and algebraic methods to confirm its location and connection to a specified point.
Understanding the Vertex of a Parabola
The vertex of a parabola given by $ h(x) = ax^2 + bx + c $ lies on its axis of symmetry. The x-coordinate of the vertex is found using the formula:
$$ x = rac{-b}{2a} $$
For the function $ h(x) = x^2 - 4x + c $:
- $ a = 1 $
- $ b = -4 $
Applying the formula:
$$ x = rac{-(-4)}{2(1)} = rac{4}{2} = 2 $$
This confirms the vertex occurs at $ x = 2 $, consistent with the given condition.
Determining the y-Coordinate of the Vertex
To find the full vertex point $ (2, h(2)) $, substitute $ x = 2 $ into the function:
$$ h(2) = (2)^2 - 4(2) + c = 4 - 8 + c = -4 + c $$
We are given that at $ x = 2 $, the function equals 3: $$ h(2) = 3 $$
Set the expression equal to 3:
$$ -4 + c = 3 $$
Solving for $ c $:
$$ c = 3 + 4 = 7 $$
Summary of the Vertex and Function Behavior
With $ c = 7 $, the full function becomes $ h(x) = x^2 - 4x + 7 $. The vertex lies at $ (2, 3) $, confirming both the axis of symmetry and the required function value at $ x = 2 $.
Final Thoughts
Finding the vertex of a parabola combines geometric intuition with algebraic precision. By using the vertex formula $ x = rac{-b}{2a} $, and substitifying to determine the constant, we accurately determine critical points that define the function’s minimum. This approach is fundamental in quadratic analysis, physics, optimization problems, and modeling real-world phenomena.
Key takeaways:
- Vertex x-coordinate: $ x = rac{-b}{2a} $
- Substitute $ x = 2 $ into $ h(x) $ to solve for $ c $ given a point on the vertex
- The vertex defines the parabola’s minimum (since $ a > 0 $) and graph symmetry
Understanding these steps empowers students and enthusiasts to confidently analyze and solve problems involving quadratic functions.
