Find the maximum or minimum value of a quadratic function

Learning Objectives Identify whether a quadratic function has a maximum or minimum value based on its leading coefficient. Find the vertex of a parabola using the algebraic formula x = -b / (2a). Find the maximum or minimum value by completing the square to write the function in vertex form f(x) = a(x - h)^2 + k. Use the first derivative test to find the critical point of a quadratic function and determine if it's a maximum or minimum. Use the second derivative test to confirm the nature of the extremum (maximum or minimum). Apply these methods to solve real-world optimization problems involving quadratic models. Interpret the vertex and the maximum/minimum value in the context of a given problem. Ever wondered how a rocket scientist calculates the peak altitude of a...

TermDefinitionExample ParabolaThe U-shaped graph of a quadratic function, f(x) = ax^2 + bx + c.The graph of y = x^2 is a parabola that opens upwards with its lowest point at (0, 0). VertexThe highest or lowest point on a parabola. The y-coordinate of the vertex represents the maximum or minimum value of the function.For y = (x - 3)^2 + 5, the vertex is at (3, 5). Maximum ValueThe greatest possible y-value of a function. For a quadratic, this occurs at the vertex of a parabola that opens downwards (when a < 0).The function f(x) = -x^2 has a maximum value of 0, which occurs at x = 0. Minimum ValueThe smallest possible y-value of a function. For a quadratic, this occurs at the vertex of a parabola that opens upwards (when a > 0).The function f(x) = x^2 + 2 has a minimum value of 2, whi...

The Vertex Formula For a quadratic function f(x) = ax^2 + bx + c, the vertex (h, k) is found by: h = -b / (2a) and k = f(h). This is the most direct algebraic method to find the coordinates of the vertex. The y-coordinate, k, is the maximum or minimum value of the function. Vertex Form of a Quadratic f(x) = a(x - h)^2 + k This form, obtained by completing the square, directly reveals the vertex (h, k). The value k is the minimum value if a > 0 and the maximum value if a < 0. The First Derivative Test for Extrema For a function f(x), find the critical points by solving f'(x) = 0. For a quadratic f(x) = ax^2 + bx + c, this is f'(x) = 2ax + b = 0. This calculus-based approach identifies the x-coordinate of the vertex. The nature of the extremum (max/min)...