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 $: - Midis
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:
📰 Wreckfest 2 Launch Date Dropped! Is This the Best Summer Launch Ever? 📰 The Moment You Tried to Avoid: Wreckfest 2 Release Date Finally Confirmed! 📰 🔥 Ruining Everything: What You Need to Know About ‘Wreck It Ralph and’! You Won’t Believe the Twist! 📰 Fortnite Just Got A Legendary Makeover Can Lara Croft Dominate The Battle Royale 📰 Fortnite Kpop Demon Hunters Unleash Epic Battle Royale Madness 📰 Fortnites New Mystery Champion Say Hello To Lara Croft Elevate Your Game Instantly 📰 Found A Keepsake Box Thats Worth Thousandsheres How It Saved A Family Legacy 📰 Found Hidden Muscles In Your Armyou Need To See Whats Beneath 📰 Fox Red Labrador Retriever Puppies Are Taking The Internet By Stormheres Why 📰 Frac2516X2 25 📰 Frac43Pi R3 Frac43Pi R 23 128Pi 📰 Fracs36Sqrt2 100 📰 Fracx2Y Fracy2Z Rightarrow X2 Z Y3 Quad Fracy2Z Fracz2X Rightarrow Y2 X Z3 📰 Free Kinemaster Logo Png Download Now Make Your Content Pop 📰 Free Kitty Coloring Pages Fun Relaxing Activity No Cat Lover Should Miss 📰 Freight Train Into Victory The Kart Leya That Set Online Street Racing Talks Ablaze 📰 From Y X And X Y Substituting X X Rightarrow X 0 Then Y 0 Only Solution Is Mathbfv Eginpmatrix 0 0 Endpmatrix But Mathbfv2 0 📰 From 1Final 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 $:
$$
c = 3 + 4 = 7
$$
