Solution: The maximum height occurs at the vertex. For $ y = -x^2 + 6x - 8 $, $ a = -1 $, $ b = 6 $, so $ x = -\frac62(-1) = 3 $. Substitute $ x = 3 $: $ y = -(3)^2 + 6(3) - 8 = -9 + 18 - 8 = 1 $. The maximum height is $ 1 $ unit. \boxed1 - Professionaloutdoormedia
Understanding the Maximum Height of a Parabola: A Step-by-Step Solution
Understanding the Maximum Height of a Parabola: A Step-by-Step Solution
When analyzing quadratic functions, one essential concept is identifying the vertex, which represents the maximum or minimum point of the parabola. In cases where the parabola opens downward (i.e., the coefficient of $x^2$ is negative), the vertex corresponds to the highest point — the maximum height.
This article walks through a clear, step-by-step solution to find the maximum value of the quadratic function $ y = -x^2 + 6x - 8 $.
Understanding the Context
Step 1: Recognize the Standard Form
The given quadratic equation is in standard form:
$$
y = ax^2 + bx + c
$$
Key Insights
Here,
- $ a = -1 $
- $ b = 6 $
- $ c = -8 $
Since $ a < 0 $, the parabola opens downward, confirming a maximum value exists at the vertex.
Step 2: Calculate the x-Coordinate of the Vertex
The x-coordinate of the vertex is found using the formula:
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$$
x = - rac{b}{2a}
$$
Substitute $ a = -1 $ and $ b = 6 $:
$$
x = - rac{6}{2(-1)} = - rac{6}{-2} = 3
$$
So, the vertex occurs at $ x = 3 $.
Step 3: Substitute to Find the Maximum y-Value
Now plug $ x = 3 $ back into the original equation to find $ y $:
$$
y = -(3)^2 + 6(3) - 8 = -9 + 18 - 8 = 1
$$
Thus, the maximum height is $ y = 1 $ unit.
