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) $:

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:

$$
x = rac{-b}{2a}
$$

$Solution: The maximum of $ P(x) = -x^2 + 4x + m $ occurs at the vertex. The $ x $-coordinate of the vertex is $ x = \frac{-b}{2a} = \frac{-4}{-2} = 2 $. Substitute $ x = 2 $ into $ P(x) $:$

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:

📰 warabimochi 📰 warbucks 📰 warbucks daddy 📰 Unlock Your Ages Secret The Ultimate Heart Rate Variability Chart You Wont Ignore 📰 Unlock Your Best Half Marathon Pacethis Chart Will Change How You Run Forever 📰 Unlock Your Best Look 10 Eye Catching Hair Color Ideas You Need Now 📰 Unlock Your Best Self With Happiness Colouring 7 Books That Transform Your Mood 📰 Unlock Your Creativity The Coolest Halloween Wallpapers You Need Now 📰 Unlock Your Gardens Full Potential Start Your Own Seed Stock Today 📰 Unlock Your Gardens Secret The Hidden Codes Youve Been Missing 📰 Unlock Your Inner Artist Coloring Pages To Make Your Heart Glow 📰 Unlock Your Inner Rockstar With These 3 Must Play Guitar Hero 3 Songs 📰 Unlock Your Inner Wizard Learn Harry Potter Drawing Like A Pro 📰 Unlock Your Inner Wizard With These Stunning Harry Potter Coloring Pages 📰 Unlock Your Jewish Heritage Discover The Ultimate Hebrew Tastatur For Perfect Typing 📰 Unlock Your Perfect Hair Type Expert Tips For Every Male Style 📰 Unlock Your Win At Happy108Online The Surprise Impact You Dont Want To Miss 📰 Unlocking Grimmjows Secrets Why Every Android Rival Is Obsessed

Final 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.