Graph each function. Determine the largest open intervals of the domain over which each function is (a) increasing or (b) decreasing. See Example 1. ƒ(x)=1/2(x-2)^2+4

Key Concepts

Function Graphing

Graphing a function involves plotting its points on a coordinate plane to visualize its behavior. For the function ƒ(x)=1/2(x-2)^2+4, this means identifying key features such as the vertex, axis of symmetry, and intercepts. Understanding how to graph quadratic functions is essential for analyzing their increasing and decreasing intervals.

Increasing and Decreasing Intervals

Increasing and decreasing intervals refer to the sections of the graph where the function's output values rise or fall as the input values increase. A function is increasing on an interval if, for any two points within that interval, the function value at the higher input is greater than at the lower input. Conversely, it is decreasing if the function value at the higher input is less.

Derivative and Critical Points

The derivative of a function provides information about its rate of change, helping to identify critical points where the function's behavior changes. By finding where the derivative is zero or undefined, we can determine potential maximums, minimums, and points of inflection. This analysis is crucial for establishing the intervals of increase and decrease for the function.