Question

The functions f(x) = (x + 3)² - 2 and g(x) = -(x - 5)^2 + 1 have been rewritten using the completing-the-square method. Apply your

knowledge of functions in vertex form, determine the vertex for each function and identify if the vertex is a minimum or a maximum and explain your reasoning.

Answer 1

The vertices of the quadratic functions f(x) and g(x) are at (-3, -2) and (5, 1) respectively, with f(x) having a minimum and g(x) a maximum due to the signs of their leading coefficients.

Step-by-step explanation:

The student is asking about determining the vertex of quadratic functions in vertex form and whether the vertex represents a minimum or maximum. For the given functions f(x) = (x + 3)² - 2 and g(x) = -(x - 5)² + 1, we can identify the vertices directly from the equations. Since both equations are already in vertex form, the vertex for f(x) is at (-3, -2) and the vertex for g(x) is at (5, 1).

The coefficient of the squared term indicates whether the parabola opens upwards or downwards. In f(x), the squared term is positive, so the parabola opens upwards, making the vertex at (-3, -2) a minimum. Conversely, in g(x), the squared term is negative, indicating the parabola opens downwards, thus the vertex at (5, 1) is a maximum.