Determine the largest open interval of the domain (a) over which the function is increasing and (b) over which it is decreasing. See Example 2. ƒ(x) = (x + 3)^2

Verified step by step guidance

1. Step 1: Identify the function type. The given function is \( f(x) = (x + 3)^2 \), which is a quadratic function. Quadratic functions are parabolas, and they have a vertex that represents either a maximum or minimum point.
2. Step 2: Determine the vertex of the parabola. The function \( f(x) = (x + 3)^2 \) is in vertex form \( f(x) = (x - h)^2 + k \), where \( h \) and \( k \) are the coordinates of the vertex. Here, the vertex is at \( (-3, 0) \).
3. Step 3: Analyze the direction of the parabola. Since the coefficient of \( x^2 \) is positive, the parabola opens upwards. This means the function decreases to the left of the vertex and increases to the right of the vertex.
4. Step 4: Determine the interval where the function is increasing. Since the function increases to the right of the vertex, the interval where the function is increasing is \((-3, \infty)\).
5. Step 5: Determine the interval where the function is decreasing. Since the function decreases to the left of the vertex, the interval where the function is decreasing is \((-\infty, -3)\).

Key Concepts

Function Behavior

Understanding how a function behaves involves analyzing its increasing and decreasing intervals. A function is increasing on an interval if, for any two points within that interval, the function's value at the second point is greater than at the first. Conversely, it is decreasing if the function's value at the second point is less than at the first. This behavior is crucial for determining the intervals of increase and decrease.

Derivative

The derivative of a function provides information about its rate of change. For a function to be increasing, its derivative must be positive, while a negative derivative indicates that the function is decreasing. By finding the derivative of the given function and analyzing its sign, we can identify the intervals where the function is increasing or decreasing.

Critical Points

Critical points occur where the derivative of a function is zero or undefined. These points are essential for determining where a function changes from increasing to decreasing or vice versa. By locating the critical points of the function ƒ(x) = (x + 3)^2, we can analyze the intervals around these points to establish where the function is increasing or decreasing.