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:

📰 guys Are Cutting Like Never Before—Discover the Hottest Asian Hair Trends! 📰 From Tight Runs to Bold Bangs: The Asian Hairstyles Guys Successfully Nailed! 📰 Asian Hairstyles for Guys: The Ultimate Guide to Turning Heads Today! 📰 Can Mario Save The Universe In The Epic Super Mario Galaxy Movie Find Out 📰 Can Mario Survive The Sequel The Extreme Secrets In Super Mario Movie 2 📰 Can Mario Win Again The Epic Sequel Of Super Mario Bros Movie 2 Shocked Fans 📰 Can One Experiment Change Everything Test Fusion Delivers The Unexpected 📰 Can One Outfit Do It All Discover The Revolutionary Sweatshirt Dress Trend Here 📰 Can One Red Suit Change Your Career Science Says Yessee How 📰 Can One Summer Smith Make Your Summer Unforgettable Discover How 📰 Can Returning Stars Deliver A Blowout Suicide Squad 2 Performance Watch What Happens 📰 Can Ryu Take On The Modern Street Fight Review Of The Ultimate Real World Battle Rig 📰 Can Someone Hear Me The Shocking Reality Of Talking To A Brick Wall You Wont Believe 3 📰 Can Street Fighter 3 Third Strike Beat Legacy Fighters Shocking Reveal Inside 📰 Can Styrofoam Balls Save Your Garden This Surprising Hack Will Astound You 📰 Can Sucrose Genshin Beat Genshins Biggest Secrets Fans Are Obsessed 📰 Can Sunshine Deia Tutt Change Your Life Discover Her Mind Blowing Story That Everyones Craving 📰 Can Super Mario Strikers Beat You Test Your Skills In This Unstoppable Online Battle

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.