Solution: The quadratic $ p(t) = -t^2 + 14t + 30 $ opens downward. The vertex occurs at $ t = -\fracb2a = -\frac142(-1) = 7 $. Substituting $ t = 7 $, $ p(7) = -(7)^2 + 14(7) + 30 = -49 + 98 + 30 = 79 $.

The Full Solution to Finding the Maximum Value of the Quadratic Function $ p(t) = -t^2 + 14t + 30 $

Understanding key features of quadratic functions is essential in algebra and real-world applications such as optimization and curve modeling. A particularly common yet insightful example involves analyzing the quadratic function $ p(t) = -t^2 + 14t + 30 $, which opens downward due to its negative leading coefficient. This analysis reveals both the vertex — the point of maximum value — and the function’s peak output.

Identifying the Direction and Vertex of the Quadratic

Understanding the Context

The given quadratic $ p(t) = -t^2 + 14t + 30 $ is in standard form $ ax^2 + bx + c $, where $ a = -1 $, $ b = 14 $, and $ c = 30 $. Because $ a < 0 $, the parabola opens downward, meaning it has a single maximum point — the vertex.

The $ t $-coordinate of the vertex is found using the formula $ t = -rac{b}{2a} $. Substituting the values:

$$
t = -rac{14}{2(-1)} = -rac{14}{-2} = 7
$$

This value, $ t = 7 $, represents the hour or moment when the quantity modeled by $ p(t) $ reaches its maximum.

$Solution: The quadratic $ p(t) = -t^2 + 14t + 30 $ opens downward. The vertex occurs at $ t = -\frac{b}{2a} = -\frac{14}{2(-1)} = 7 $. Substituting $ t = 7 $, $ p(7) = -(7)^2 + 14(7) + 30 = -49 + 98 + 30 = 79 $.$

Key Insights

Calculating the Maximum Value

To find the actual maximum value of $ p(t) $, substitute $ t = 7 $ back into the original equation:

$$
p(7) = -(7)^2 + 14(7) + 30 = -49 + 98 + 30 = 79
$$

Thus, the maximum value of the function is $ 79 $ at $ t = 7 $. This tells us that when $ t = 7 $, the system achieves its peak performance — whether modeling height, revenue, distance, or any real-world behavior described by this quadratic.

Summary

🔗 Related Articles You Might Like:

📰 ar glasses 📰 gps tracker for kids 📰 how to add screen record on iphone 📰 Apocalypse Horses Undead Nightmare 📰 Report Reveals Clues For Connections Last Update 2026 📰 Data Shows High Speed Internet Access Satellite And It S Going Viral 📰 Todays The Young And The Restless 2391260 📰 Steam Msfs 2024 📰 Aspire Credit Card Login 📰 Crazeygames 📰 Recursively Remove Directory Linux 📰 Certificate Of Deposit Ladder 📰 Sdc Swingers Secrets The Hidden Rules That Will Change Everything 8755973 📰 Shock Moment Watch Skinamarink And It Raises Alarms 📰 Bank Of America Checks Designs 📰 From Thought Leadership To Hiring Excellence Oracles Secret To Recruiting The Best Tech Talent 8586300 📰 This Simple Hack Let Me Make Fire Sticks Like A Pro In Minecraft 8339602 📰 Could These Beans Be The Hidden Treasure Ruining Your Dinner 8667496

Final Thoughts

Function: $ p(t) = -t^2 + 14t + 30 $ opens downward ($ a < 0 $)
Vertex occurs at $ t = -rac{b}{2a} = 7 $
Maximum value is $ p(7) = 79 $

Knowing how to locate and compute the vertex is vital for solving optimization problems efficiently. Whether in physics, economics, or engineering, identifying such key points allows for precise modeling and informed decision-making — making the vertex a cornerstone of quadratic analysis.