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:

📰 Eória Of Hello Kitty & Blooms: The Sweetest Bouquet You Won’t Want to Miss! 📰 Click to See the Hello Kitty Flower Arrangement That’s Taking Social Media by Storm! 📰 Hello Kitty Magnetic Flip Phone Secret: Retro Features You NEED! 📰 The Dark Seduction Of Love Tyrant Whats Driving His Powerful Fanbase 📰 The Dark Side Of Luffy Explained You Wont Believe Why Its Misunderstood 📰 The Dark Side Of Marx Kirby Shocking Facts That Will Blow Your Mind 📰 The Dark Truth Behind Loup Garoul Mind Blowing Facts No One Talks About 📰 The Darkest Marvel Villains That Will Haunt Your Dreams Click To Discover Their Full Story 📰 The Darkest Secret Behind Manga Made In The Abyss Thats Taking Fans Deranged 📰 The Darkest Secrets Lord Voldemorts Unbelievable Rise And Fall In Harry Potter 📰 The Darkest Truth About Lord Voldemorts Forbidden Tom You Wont Believe What Blazed Through His Soul 📰 The Data Analyst Compares Two Interventions One Reduced Er Transfers By 25 The Other By 16 Which Led To A Greater Reduction And What Was The Difference In Decimal Form 📰 The Data Analyst Finds That Reducing Patient Readmission Rates In 8 Clinics Led To An Average Drop From 18 To 12 If Each Clinic Serves 250 Patients Annually How Many Fewer Readmissions Occurred Citywide Due To This Improvement 📰 The Deepest Hillsong Worship Lyrics That Will Make You Praywatch These What A Beautiful Name Lyrics Fire Up Your Soul 📰 The Definitive Breakdown Breathe In Every Powerful Line Of We Belong Together Lyrics 📰 The Definitive Chronological Ride All Marvel Movies Explained In Order Youve Been Waiting For 📰 The Delicious Truth About Louis Lanes Hidden Career You Didnt Know 📰 The Denominator Is

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.