The function \( R(t) = 3t^2 - 12t + 15 \) is a quadratic in standard form \( at^2 + bt + c \), with \( a = 3 > 0 \), so it opens upwards and has a minimum at its vertex.

["# Understanding the Quadratic Function ( R(t) = 3t^2 - 12t + 15 ): Vertex, Shape, and Minimum Value", "Quadratic functions are fundamental in algebra, offering powerful insights into parabolas, optimization, and real-world modeling. One such quadratic function is ( R(t) = 3t^2 - 12t + 15 ), which admits a clear standard form ( at^2 + bt + c ). In this article, we explore the key features of this function—its coefficients, direction, vertex, and the existence of a minimum value—so you can better understand and apply quadratic reasoning in mathematics and beyond.", "## What is a Quadratic in Standard Form?", "A quadratic function takes the form:", "[\nR(t) = at^2 + bt + c\n]", "where ( a ), ( b ), and ( c ) are constants and ( a <br/>\neq 0 ). In our function ( R(t) = 3t^2 - 12t + 15 ), comparing terms gives:\n- ( a = 3 )\n- ( b = -12 )\n- ( c = 15 )", "This standard form allows us to identify core properties like direction of opening and location of the vertex — key to graphing and analyzing the function.", "## Determining the Parabola’s Direction", "The coefficient ( a ) governs the concavity and direction of the parabola:", "- If ( a > 0 ), the parabola opens upward\n- If ( a < 0 ), it opens downward", "Here, ( a = 3 > 0 ), so the parabola opens upward, meaning the function has a minimum value at its vertex. This upward curvature is crucial for applications where modeling a minimum quantity makes sense—such as cost minimization or optimization problems.", "## Finding the Vertex – Location of the Minimum", "The vertex of a parabola given by ( R(t) = at^2 + bt + c ) lies at the point:", "[\nt_v = -\frac{b}{2a}\n]", "Plugging in ( a = 3 ) and ( b = -12 ):", "[\nt_v = -\frac{-12}{2 \cdot 3} = \frac{12}{6} = 2\n]", "So, the minimum value occurs at ( t = 2 ). Since the parabola opens upward, this gives the lowest point on the graph.", "## Evaluating the Minimum Value of the Function", "To find the minimum value ( R(t_v) = R(2) ), substitute ( t = 2 ) into the function:", "[\nR(2) = 3(2)^2 - 12(2) + 15 = 3 \cdot 4 - 24 + 15 = 12 - 24 + 15 = 3\n]", "Thus, the minimum value of the function is 3, achieved when ( t = 2 ).", "## Geometric Interpretation: Symmetry and the Vertex", "Because the parabola is symmetric, points equidistant from the vertex ( t = 2 ) have equal function values. For example, calculate ( R(0) ) and ( R(4) ):", "[\nR(0) = 3(0)^2 - 12(0) + 15 = 15\n]\n[\nR(4) = 3(4)^2 - 12(4) + 15 = 48 - 48 + 15 = 15\n]", "Indeed, ( R(0) = R(4) ), confirming symmetry about the line ( t = 2 ). This symmetry underscodes the function’s elegant structure.", "## Real-World Implications", "Quadratic functions like ( R(t) ) model many natural and engineered phenomena involving minimization or maximization—such as projectile motion, optimization of profit and cost, or design efficiency. Knowing the vertex allows decision-makers to pinpoint the optimal input (like dosage, time, or production level) yielding the best outcome (minimum cost, maximum yield, etc.).", "## Summary", "- The function ( R(t) = 3t^2 - 12t + 15 ) is a quadratic in standard form with ( a = 3 > 0 ), so it opens upward.\n- The vertex, located at ( t = 2 ), gives the minimum value of ( R(t) = 3 ).\n- This minimum reflects the lowest point on the parabola, crucial for optimization applications.\n- The parabola’s symmetry about ( t = 2 ) ensures balanced behavior around the minimum.", "Understanding these features empowers students and practitioners to analyze, graph, and apply quadratic functions with confidence across mathematics and real-life contexts.", "---", "Keywords: quadratic function, ( R(t) = 3t^2 - 12t + 15 ), standard form ( at^2 + bt + c ), parabola opens upwards, vertex, minimum value, upward opening quadratic, vertex formula, quadratic optimization."]