Suppose That The Function G Is Defined As Follows

The function g is definedas follows:

g(x) = 3x² - 4x + 7

This quadratic function, with its standard form ax² + bx + c, reveals a parabolic curve opening upwards due to the positive coefficient of x² (a=3). The vertex, representing the function's minimum point, occurs at x = -b/(2a) = -(-4)/(23) = 4/6 = 2/3. Substituting x=2/3 yields g(2/3) = 3(4/9) - 4*(2/3) + 7 = 12/9 - 8/3 + 7 = 4/3 - 8/3 + 7 = -4/3 + 21/3 = 17/3 ≈ 5.67. Thus, the vertex is at (2/3, 17/3), and the y-intercept is g(0) = 7.

Understanding the behavior of g(x) requires analyzing its derivative, g'(x) = 6x - 4. Setting this equal to zero identifies critical points: 6x - 4 = 0 → x = 4/6 = 2/3. The second derivative, g''(x) = 6, is positive, confirming this critical point as a local minimum. For x < 2/3, g'(x) < 0, indicating g is decreasing; for x > 2/3, g'(x) > 0, indicating g is increasing. This confirms the vertex is the minimum point.

The range of g(x) is [17/3, ∞), as the parabola opens upwards with no upper bound. The function is defined for all real numbers, with domain (-∞, ∞). Its symmetry axis is the vertical line x = 2/3. Evaluating specific points: g(0) = 7, g(1) = 3-4+7=6, g(2) = 12-8+7=11, g(3) = 27-12+7=22, and g(-1) = 3+4+7=14. These values illustrate the function's increasing behavior post-vertex and decreasing behavior pre-vertex.

This function's properties make it useful for modeling scenarios involving minimum thresholds or growth bounded below. Its predictable curvature and vertex location provide clear insights for optimization problems. Understanding its algebraic and calculus-based characteristics allows for efficient analysis of its behavior across any interval.