Solution: The vertex form of a parabola $ y = ax^2 + bx + c $ has its vertex at $ x = -\fracb2a $. Here, $ a = -2 $, $ b = 8 $, so $ x = -\frac82(-2) = 2 $. Substituting $ x = 2 $ into the equation: $ y = -2(2)^2 + 8(2) - 5 = -8 + 16 - 5 = 3 $. Thus, the vertex is at $ (2, 3) $. \boxed(2, 3)

Mastering Parabolas: Finding the Vertex Using Vertex Form

Understanding the shape and position of a parabola is essential in algebra and calculus. One key feature is the vertex — the highest or lowest point depending on the parabola’s direction. For quadratic equations in standard form $ y = ax^2 + bx + c $, calculating the vertex coordinates has a simple, efficient solution known as the vertex formula.

What Is the Vertex of a Parabola?

Understanding the Context

The vertex of a parabola described by $ y = ax^2 + bx + c $ occurs at
$$
x = -rac{b}{2a}
$$
This formula is derived from completing the square and reveals the axis of symmetry. Once the $ x $-coordinate is found, substituting it back into the equation gives the $ y $-coordinate, pinpointing the exact vertex.

Applying the Vertex Formula

Let’s apply this method step-by-step using a concrete example.

Given the quadratic function:
$$
y = -2x^2 + 8x - 5
$$
Here, the coefficients are $ a = -2 $, $ b = 8 $, and $ c = -5 $.

$Solution: The vertex form of a parabola $ y = ax^2 + bx + c $ has its vertex at $ x = -\frac{b}{2a} $. Here, $ a = -2 $, $ b = 8 $, so $ x = -\frac{8}{2(-2)} = 2 $. Substituting $ x = 2 $ into the equation: $ y = -2(2)^2 + 8(2) - 5 = -8 + 16 - 5 = 3 $. Thus, the vertex is at $ (2, 3) $. \boxed{(2, 3)}$

Key Insights

Find the $ x $-coordinate of the vertex:

$$
x = -rac{b}{2a} = -rac{8}{2(-2)} = -rac{8}{-4} = 2
$$

Determine the $ y $-coordinate by substituting $ x = 2 $:

$$
y = -2(2)^2 + 8(2) - 5 = -2(4) + 16 - 5 = -8 + 16 - 5 = 3
$$

So, the vertex of this parabola is at $ (2, 3) $.

🔗 Related Articles You Might Like:

📰 Macfan Control 📰 Free Adobe for Mac 📰 Can You Get Sims on Macbook 📰 Three Branches Of Federal Government 791533 📰 Best Sandbox Games Steam 1679640 📰 The Surprising Power Of Copper Stock You Cant Ignore Right Now 5323181 📰 Tata Motors Stock Explodescan You Beat The Huge Share Rate Surge 4751593 📰 Youll Never Guess Dogs Can Eat Quinoa The Surprising Truth 715520 📰 Last Minute Vacation Trips 8824373 📰 5Explode Tata Motors Stock Breakout See What Investors Are Working On Now 2145259 📰 Thai Solar Calendar 3720288 📰 You Wont Believe What Happens When You Try Comt Like Never Before 1320022 📰 Sdm Stock Price Soaringwhat Investors Are Missing Could Be Massive Profits 6841323 📰 The Hidden Truth About The Java Logo That Developers Are Obsessed With 7732311 📰 Pumping Iron Bodybuilders 2164605 📰 Apple Manhattan Village Manhattan Beach Ca 2667146 📰 American Eagle Job Search 6687542 📰 Galveston County News Obituary 2962678

Final Thoughts

Why This Method Matters

Using $ x = -rac{b}{2a} $ avoids the labor of completing the square manually and provides a direct route to the vertex. This efficiency is especially useful when analyzing symmetry, gradients, or optimization problems involving parabolas.

Final Vertex Summary

Vertex: $ (2, 3) $
Axis of Symmetry: $ x = 2 $

Mastering this vertex form and formula empowers students and professionals alike in algebra, physics, and engineering — where parabolic motion and curve modeling are key.

Try it yourself: Try any quadratic in standard form — apply $ x = -rac{b}{2a} $ to find the vertex quickly using the vertex formula!

oxed{(2, 3)}