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) $: - RTA
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:
📰 powerball numbers november 1st 📰 powerball tonigt 📰 who do the cowboys play this sunday 📰 Turnberry Golf Club 2397370 📰 You Wont Believe What Jjk R34 Did To The Fanbasespoiler Its Bombs 6393757 📰 Wells Fargo Checking Account Features 7804530 📰 Can Percento Increase Your Income Faster Than You Think Find Out Now 8356779 📰 South Park Longer And Uncut 1997142 📰 All Fortnite Skins 2534235 📰 You Wont Believe What Happened In Madagascar Escape 2 Africa Strike Back 4070543 📰 Living Room Theaters Indianapolis 2500752 📰 This Mutant Face Makeover Is Revolutionagree Mutantsnew 4198486 📰 X Men 3 Last Stand Revealed The Shocking Twists That Redefined The Universe 1829350 📰 Apt Lyrics 3497799 📰 Aberdeen Township 3613378 📰 Indiana Salary Database 1926997 📰 Swear This Gritona Is The Harshest Tequila Youve Ever Tasted 2938904 📰 How A 2 Car Garage Size Maximizes Your Home The Secret You Need To Know 5651337Final 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.
