The function is a quadratic in the form \( h(t) = at^2 + bt + c \), with \( a = -5 \), \( b = 150 \), and \( c = 100 \).

["Exploring the Quadratic Function ( h(t) = -5t^2 + 150t + 100 ): Behavior, Maximum, and Applications", "A quadratic function is defined by the general form ( h(t) = at^2 + bt + c ), where ( a ), ( b ), and ( c ) are constants, and ( a <br/>\neq 0 ). In this article, we analyze the specific quadratic function ( h(t) = -5t^2 + 150t + 100 ), focusing on its mathematical properties, graphical behavior, maximum height, and practical applications.", "### Understanding the Structure of the Quadratic Function", "The function ( h(t) = -5t^2 + 150t + 100 ) has:", "- ( a = -5 ): Determines the concavity and direction of the parabola\n- ( b = 150 ): Influences the vertex’s horizontal position\n- ( c = 100 ): Sets the y-intercept", "Because ( a = -5 ) is negative, the parabola opens downward, meaning it has a single maximum point—a vertex that represents the highest value of the function.", "### Finding the Vertex: Maximum Height of the Function", "For any quadratic ( h(t) = at^2 + bt + c ), the time ( t ) at which the maximum (or minimum) occurs is given by the vertex formula:\n[\nt = -\frac{b}{2a}\n]", "Substituting ( a = -5 ) and ( b = 150 ):\n[\nt = -\frac{150}{2 \ imes (-5)} = -\frac{150}{-10} = 15\n]", "The maximum height occurs at ( t = 15 ). To find the corresponding maximum value ( h_{\ ext{max}} ), substitute ( t = 15 ) into the function:", "[\nh(15) = -5(15)^2 + 150(15) + 100 = -5(225) + 2250 + 100 = -1125 + 2250 + 100 = 1225\n]", "Thus, the function reaches its maximum value of 1225 at ( t = 15 ).", "### Graphical Behavior and Key Features", "The graph of ( h(t) ) is a downward-opening parabola. Key features include:", "- Vertex: Point ( (15, 1225) ), the peak of the curve\n- Y-intercept: When ( t = 0 ), ( h(0) = 100 )\n- X-intercepts (Roots): Solutions to ( -5t^2 + 150t + 100 = 0 ); solved using the quadratic formula:\n[\nt = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-150 \pm \sqrt{150^2 - 4(-5)(100)}}{2(-5)} = \frac{-150 \pm \sqrt{22500 + 2000}}{-10} = \frac{-150 \pm \sqrt{24500}}{-10}\n]", "Simplifying:\n[\n\sqrt{24500} = \sqrt{100 \ imes 245} = 10\sqrt{245} \approx 10 \ imes 15.65 = 156.5\n]\n[\nt = \frac{-150 \pm 156.5}{-10}\n]", "Calculating both roots:\n1. ( t = \frac{-150 + 156.5}{-10} = \frac{6.5}{-10} = -0.65 )\n2. ( t = \frac{-150 - 156.5}{-10} = \frac{-306.5}{-10} = 30.65 )", "So, the parabola crosses the horizontal axis at approximately ( t \approx -0.65 ) (before ( t = 0 ), not meaningful in most physical contexts) and ( t \approx 30.65 ).", "### Real-W世界 Applications of This Quadratic Model", "Quadratic functions modeling height over time are common in physics, particularly in projectile motion where objects are launched into the air under gravity (ignoring air resistance). Here, ( h(t) ) represents height (in meters, for example) as a function of time (in seconds), with ( t = 0 ) at launch.", "- The negative coefficient ( a = -5 ) simulates gravitational acceleration (magnitude approximated as 10 m/s² scaled for simplicity).\n- The positive ( b = 150 ) reflects a strong initial velocity ("bang" or upward push) of 150 m/s (scaled).\n- The maximum height ( h_{\ ext{max}} = 1225 ) meters indicates a high arc, and the duration until landing spans about 61 seconds (from ~0.65s to 30.65s).", "This functional form helps engineers, educators, and scientists predict trajectories, optimize launch angles, and design safer, more efficient motion systems.", "### Conclusion", "The quadratic function ( h(t) = -5t^2 + 150t + 100 ) offers a rich model of upward motion with a clear maximum, shaped by its parameters: downward curvature, strong initial push, high peak, and predictable descent. Whether applied in physics, architecture, or data analysis, understanding such quadratics strengthens both mathematical insight and real-world problem-solving skills.", "---", "Keywords: quadratic function, ( h(t) = at^2 + bt + c ), maximum height, vertex formula, projectile motion, physics applications, concave down parabola, quadratic modeling.\nMeta Description: Explore the quadratic ( h(t) = -5t^2 + 150t + 100 ), its properties, vertex at ( t = 15 ), maximum height 1225, and real-world projectile motion applications. Learn how this function models upward trajectories and peak performance."]