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

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:

$$
y = ax^2 + bx + c
$$

$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}$

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:

🔗 Related Articles You Might Like:

📰 paramount nickelodeon shows removed permanently or temporarily 📰 clipper for nose hair 📰 2024 uk election 📰 Soil Ph Test Kit You Wont Believe Changes Your Garden Results Forever 1639664 📰 Soul Men And The Soul Man Truth Samuel Jackson Shocked Fans With In His Warning 4804440 📰 The Hottest 3D Games Of 2024 Are Heregame Changers You Cant Ignore 4305134 📰 40 2 4028080 4366476 📰 Yellowstone Lodging 2924946 📰 South Carolinas Pulse Right Nowcan You Keep Up 8378757 📰 Only The Chosen See It The Haunting Rebirth Of Souls In Bleach That Will Blow Your Mind 345177 📰 Foot Cream For Dry Feet 4708361 📰 Water Melon Game Hacked Can You Beat The Record Click To Find Out 631919 📰 Subsidio 6441589 📰 Hercules The Rock 4054738 📰 Kim Scott Mathers 4576673 📰 Light Colors With A Pop Light Colorful Art That Steals Every Eye 6049039 📰 Prequalify For Mortgage Loan 7932899 📰 Husband Wendy Williams 6030064

Final Thoughts

$$
x = -rac{b}{2a}
$$

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.