Solution: The vertex of a parabola $ h(x) = x^2 - 4x + c $ occurs at $ x = \frac-(-4)2(1) = 2 $, which matches the given condition. Now substitute $ x = 2 $ and set $ h(2) = 3 $: - Aurero
Finding the Vertex of the Parabola $ h(x) = x^2 - 4x + c $: A Step-by-Step Solution
Finding the Vertex of the Parabola $ h(x) = x^2 - 4x + c $: A Step-by-Step Solution
When analyzing quadratic functions, identifying the vertex is essential for understanding the graph’s shape and behavior. In this article, we explore how to find the vertex of the parabola defined by $ h(x) = x^2 - 4x + c $, using calculus and algebraic methods to confirm its location and connection to a specified point.
Understanding the Context
Understanding the Vertex of a Parabola
The vertex of a parabola given by $ h(x) = ax^2 + bx + c $ lies on its axis of symmetry. The x-coordinate of the vertex is found using the formula:
$$
x = rac{-b}{2a}
$$
For the function $ h(x) = x^2 - 4x + c $:
Key Insights
- $ a = 1 $
- $ b = -4 $
Applying the formula:
$$
x = rac{-(-4)}{2(1)} = rac{4}{2} = 2
$$
This confirms the vertex occurs at $ x = 2 $, consistent with the given condition.
🔗 Related Articles You Might Like:
📰 Why Every Pro Swears By 2SOC—Click to Discover the Game-Changer! 📰 They Made a ‘2nd WW Epic—Witness History Like Never Before (You Won’t Believe the Secrets It Reveals!) 📰 Two WW Movies That Changing Everything—This Sudden Revelation Shocked the World! 📰 The Horror Inside Two Hitmen Unleash Chaosyou Must See This 📰 The Horror Intensifies Second Season Secrets Everyones Raving About You Need To Watch 📰 The Hotnew 2025 Video Games That Are Taking The World By Stormdont Miss Out 📰 The Hottest 2024 Movies Breaking Streams Theatersyou Need To Watch Now 📰 The Iconic 1962 Chevy Impala Why This Muscle Car Changed Automotive History Forever 📰 The Iconic 1967 Dodge Charger Why This Classic Is Cooler Than Ever 📰 The Iconic 1977 Hoodie You Wont Stop Looking At This Vintage Piece Is Timeless 📰 The Incredible Truth About 2Nd Jumanji You Didnt Know 📰 The Incredible Value Hidden In The 1944 Wheat Penny You Might Still Have 📰 The Infamous 450 Sutter Street What Hidden Secrets Lurk Behind Those Doors 📰 The Ingenious Strategy Behind 10000 6 That No One Talks Abouta Game Changer 📰 The Insane Upgrade Trap That Made The 1999 Toyota Corolla A Hidden Collectors Gem 📰 The Largest Bra Size That Still Looks Flawless Heres Your 32B Cup Secret 📰 The Legend Of The 1St Gen Cummins Why This Engine Still Rules The Road 📰 The Legendary 1970 Ford Truck Restored Like Never Beforewatch How It Stuns CollectorsFinal Thoughts
Determining the y-Coordinate of the Vertex
To find the full vertex point $ (2, h(2)) $, substitute $ x = 2 $ into the function:
$$
h(2) = (2)^2 - 4(2) + c = 4 - 8 + c = -4 + c
$$
We are given that at $ x = 2 $, the function equals 3:
$$
h(2) = 3
$$
Set the expression equal to 3:
$$
-4 + c = 3
$$
Solving for $ c $:
$$
c = 3 + 4 = 7
$$
