Solution: The maximum of $ P(x) = -x^2 + 4x + m $ occurs at the vertex. The $ x $-coordinate of the vertex is $ x = \frac-b2a = \frac-4-2 = 2 $. Substitute $ x = 2 $ into $ P(x) $: - Deep Underground Poetry
Understanding the Maximum of the Quadratic Function $ P(x) = -x^2 + 4x + m $
Understanding the Maximum of the Quadratic Function $ P(x) = -x^2 + 4x + m $
When analyzing quadratic functions in the form $ P(x) = ax^2 + bx + c $, one of the most important concepts is identifying where the function reaches its maximum or minimum. In this case, we examine the downward-opening parabola defined by:
$$
P(x) = -x^2 + 4x + m
$$
Understanding the Context
Here, the coefficient $ a = -1 $, $ b = 4 $, and $ c = m $. Since $ a < 0 $, the parabola opens downward, meaning it has a maximum value at its vertex.
Finding the x-Coordinate of the Vertex
The $ x $-coordinate of the vertex of any quadratic function is given by the formula:
$$
x = rac{-b}{2a}
$$
Image Gallery
Key Insights
Substituting $ a = -1 $ and $ b = 4 $:
$$
x = rac{-4}{2(-1)} = rac{-4}{-2} = 2
$$
So, the vertex occurs at $ x = 2 $, which is the point where the function $ P(x) $ reaches its maximum value.
Evaluating the Maximum Value by Substituting $ x = 2 $
To find the actual maximum value of $ P(x) $, substitute $ x = 2 $ into the expression:
🔗 Related Articles You Might Like:
📰 Rates Bank of America 📰 Bank of America Business Travel Rewards 📰 Bank of America Earnings Report 📰 Dexter Seasons 2571855 📰 S And P 500 Premarket 1384302 📰 Golf Visors 1103792 📰 Youve Asked What Is Crypto Now Refer To This Before Its Too Late 1606863 📰 How To Embed A Document In Word 4233378 📰 Wells Fargo Gift Cards 8982982 📰 Californias Skin Revolution Begins Hereunlock The Secrets No One Talks About 2693864 📰 Dosbox X Download 646641 📰 Srpfcu Explained The Shocking Truth About This Underused 5886625 📰 However To Avoid Issue Re Express 9257474 📰 See What Implantation Blood Looks Likeshocking Photos You Cant Look Away From 5296500 📰 Heloc Fixed Rates 5716283 📰 Hotels In Newport Beach 8078396 📰 Fraca 2D A D A A D A 2D5 A 22 9921684 📰 Speed Is Calculated As Distance Divided By Time 5920087Final Thoughts
$$
P(2) = -(2)^2 + 4(2) + m = -4 + 8 + m = 4 + m
$$
Thus, the maximum value of $ P(x) $ is $ 4 + m $, occurring at $ x = 2 $.
Key Takeaways
- The vertex of $ P(x) = -x^2 + 4x + m $ is at $ x = 2 $, the x-coordinate where the maximum occurs.
- Evaluating the function at $ x = 2 $ yields the peak value: $ P(2) = 4 + m $.
- Understanding the vertex form helps students and learners determine key features like maximums, minima, and symmetry in quadratic functions.
This insight is crucial not only for solving optimization problems but also for graphing and interpreting real-world scenarios modeled by quadratic functions.
By recognizing that the maximum of $ P(x) $ occurs at $ x = 2 $, and computing $ P(2) = 4 + m $, you gain a powerful tool for analyzing and visualizing quadratic behavior.
