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 $: - Coaching Toolbox
Finding the Vertex of the Parabola $ h(x) = x^2 - 4x + c $: A Step-by-Step Solution
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 Context
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 $:
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Key Insights
- $ 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.
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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 $:
$$
c = 3 + 4 = 7
$$
