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

Verified step by step guidance

1. Identify the given function: \(f(x) = \frac{1}{2}(x-2)^2 + 4\). This is a quadratic function in vertex form, where the vertex is at \((2, 4)\).
2. Find the first derivative \(f'(x)\) to determine where the function is increasing or decreasing. Use the power rule and chain rule: \(f'(x) = \frac{1}{2} \cdot 2(x-2) = (x-2)\).
3. Set the derivative equal to zero to find critical points: \(f'(x) = 0 \Rightarrow x - 2 = 0 \Rightarrow x = 2\). This is where the function changes from increasing to decreasing or vice versa.
4. Analyze the sign of \(f'(x)\) on intervals determined by the critical point \(x=2\): For \(x < 2\), \(f'(x) = x-2\) is negative, so the function is decreasing on \((-\infty, 2)\). For \(x > 2\), \(f'(x)\) is positive, so the function is increasing on \((2, \infty)\).
5. Summarize the domain intervals: The function is decreasing on the largest open interval \((-\infty, 2)\) and increasing on the largest open interval \((2, \infty)\). The vertex at \(x=2\) is the minimum point.

Key Concepts

Function Graphing

Graphing a function involves plotting points that satisfy the function's equation to visualize its shape. For quadratic functions like ƒ(x) = 1/2(x-2)^2 + 4, the graph is a parabola, which helps identify key features such as vertex, axis of symmetry, and general behavior.

Domain and Intervals

The domain of a function is the set of all possible input values (x-values). Open intervals are continuous subsets of the domain where the function behaves in a specific way, such as increasing or decreasing, without including the endpoints.

Increasing and Decreasing Functions

A function is increasing on an interval if its output values rise as x increases, and decreasing if its output values fall. For quadratic functions, these intervals are determined by the vertex: the function decreases before the vertex and increases after it.