Solution: The maximum height occurs at the vertex. For $ y = -x^2 + 6x - 8 $, $ a = -1 $, $ b = 6 $, so $ x = -\frac{6}{2(-1)} = 3 $. Substitute $ x = 3 $: $ y = -(3)^2 + 6(3) - 8 = -9 + 18 - 8 = 1 $. The maximum height is $ 1 $ unit. \boxed{1}
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 $.
Step 1: Recognize the Standard Form
The given quadratic equation is in standard form:
$$ y = ax^2 + bx + c $$
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:
$$ x = - rac{b}{2a} $$
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.
Final Answer: oxed{1}
Summary The maximum height of the quadratic function $ y = -x^2 + 6x - 8 $ occurs at $ x = 3 $, and substituting this value yields a maximum y-value of 1. Understanding this vertex concept is crucial in functions modeling projectile motion, profit optimization, and other real-world applications involving parabolic trends.
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