114. Parabolas
b. When is the parabola concave up? Concave down?

Verified step by step guidance

1. To determine when a parabola is concave up or concave down, we need to look at the second derivative of its equation. A parabola is typically given by the quadratic function \( f(x) = ax^2 + bx + c \).
2. The second derivative of the function \( f(x) = ax^2 + bx + c \) is \( f''(x) = 2a \). This is a constant value, meaning it does not depend on \( x \).
3. A parabola is concave up when the second derivative is positive. Therefore, if \( 2a > 0 \), the parabola is concave up. This simplifies to \( a > 0 \).
4. Conversely, a parabola is concave down when the second derivative is negative. Therefore, if \( 2a < 0 \), the parabola is concave down. This simplifies to \( a < 0 \).
5. In summary, the concavity of a parabola is determined by the sign of the coefficient \( a \) in the quadratic function. If \( a > 0 \), the parabola is concave up, and if \( a < 0 \), it is concave down.

Key Concepts

Parabola

A parabola is a U-shaped curve that can open upwards or downwards, defined by a quadratic function of the form y = ax^2 + bx + c. The direction in which the parabola opens is determined by the sign of the coefficient 'a'. If 'a' is positive, the parabola opens upwards; if 'a' is negative, it opens downwards.

Concavity

Concavity refers to the direction in which a curve bends. A function is concave up if it bends upwards like a cup, and concave down if it bends downwards like a cap. For a parabola, concavity is determined by the sign of the leading coefficient 'a' in the quadratic equation: positive 'a' indicates concave up, while negative 'a' indicates concave down.

Second Derivative Test

The second derivative test is used to determine the concavity of a function. For a quadratic function y = ax^2 + bx + c, the second derivative is a constant, 2a. If 2a > 0, the function is concave up, and if 2a < 0, it is concave down. This test helps in identifying the nature of the parabola's curvature.