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 $: - Dachbleche24
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:
📰 A social impact strategist is organizing a technology education initiative where participants can choose from 4 coding workshops, 6 design workshops, and 5 entrepreneurship workshops. If a participant randomly selects 3 workshops, what is the probability that they choose one from each category? 📰 To find the probability of selecting one workshop from each category, we calculate the number of favorable outcomes and divide by the total number of outcomes. 📰 First, calculate the total number of ways to choose 3 workshops from 15: 📰 The Hidden Danger Lurking In Every Pu Leather Product Inside 📰 The Hidden Danger No One Talks About When It Comes To Pig Penis 📰 The Hidden Danger Only Dog Owners Should Know About Before Using Proviable 📰 The Hidden Dangers Lurking Behind Every Power Line You Ignore 📰 The Hidden Dangers Lurking In Every Peat Bale Youre Ignoring 📰 The Hidden Dangers Lurking In Your Preferred Homecare Choices 📰 The Hidden Dangers Of Wearing Patent Leathershocking Reasons You Must Read 📰 The Hidden Depths Of Pistons No One Talks About 📰 The Hidden Design Perfection Hiding Inside Your Front Door Every Time 📰 The Hidden Drain That Ruins Every Pool Slide Dreamyou Wont Believe What It Really Does 📰 The Hidden Empire Behind Piotr Severezzas Rise To Fame 📰 The Hidden Evolution Of Oshawott What You Were Never Told About Its Genesis 📰 The Hidden Fallout Of Squishing Your Macs Screen Beyond Repair 📰 The Hidden Fault Lurking In Your Engine Code P0304 Will Shock You 📰 The Hidden Feature That Makes This Portable Lifter UnstoppableFinal 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 $:
$$
c = 3 + 4 = 7
$$
