Problem 9 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

(a) Use your knowledge of shifting, flipping, and stretching to graph the function \(f(x)=-2|x-2|+4\) (b) At what value of \(x\) does \(f(x)\) attain its maximum value? At this point, what is \(f^{\prime}(x) ?\) (c) Does \(f(x)\) have a minimum value? (d) Where on the interval \(3 \leq x \leq 8\) does \(f\) take on its maximum value? Its minimum value?

Step-by-Step Solution

Answer

(a) The function is a mirrored and stretched version of the base absolute value function shifted to the right by 2 units and up by 4 units. (b) The function reaches its maximum value at \(x = 2\), at this point, \(f'(x)\) is undefined. (c) The function does not have a minimum value. (d) On the interval [3, 8], the function reaches its maximum value at \(x = 3\) and its minimum value at \(x = 8\).

Step 1: Understanding the function

First, understand the function \(f(x)=-2|x-2|+4\). It's a modification of the basic absolute value function. The number next to \(x\), '2', shifts the graph to the right by '2'. The '4' outside the function shifts the graph up by '4'. The '\(-2\)' that multiplies the function flips it upside down and stretches it by a factor of '2'.

Step 2: Graphing the function

Graph the function \(f(x)=-2|x-2|+4\). The vertex of this absolute value function is at \((2,4)\). The function stays constant until \(x = 2\), at which point it decreases linearly due to the negative coefficient of absolute \(x\).

Step 3: Finding the maximum value

Since \(f(x)\) is an absolute value function that opens downwards, it reaches its maximum at its vertex. The maximum value of \(f(x)\) occurs at \(x = 2\). At this point the derivative \(f'(x)\) is undefined as absolute functions have a corner at the vertex which makes the derivative non-existent.

Step 4: Checking for a minimum value

Since the function opens downwards and is defined for all real numbers, it continuously decreases as we move away from \(x = 2\). Therefore, the function doesn't have a minimum value.

Step 5: Evaluating maximum and minimum on the interval [3, 8]

On the interval \(3 \leq x \leq 8\), \(f(x)\) decreases linearly. Hence, it reaches its maximum at \(x = 3\) (at the start of the interval) and its minimum at \(x = 8\) (at the end of the interval).

Key Concepts

Transformation of Functions

Understanding how to graph functions is a crucial skill in algebra, and graphing transformations is like being an artist with a set of tools at your disposal. When you encounter a function like f(x) = -2|x - 2| + 4, think of it as a basic absolute value function that has undergone several transformations.

Each element like the '2' in 'x - 2' or the '4' in '+ 4' indicates a shift. Specifically, the function is moved 2 units to the right and 4 units up. The negative sign before the 2 indicates a reflection across the x-axis, which flips the graph upside down. Additionally, the '2' is a vertical stretch that makes the function steeper. When graphing, start with the basic shape of the absolute value graph, apply these transformations step-by-step, and you will end up with the desired function's graph.

Finding Maximum Value of a Function

Many real-world situations, like finding the optimal profit or the highest point of a projectile, require us to determine the maximum value of a function. When dealing with absolute value functions, such as f(x) = -2|x - 2| + 4, the maximum value is found at the vertex.

For the given function, the vertex is located at (2, 4). This is the highest point of the graph because the function opens downwards due to the negative coefficient. It's important to remember that the maximum value is not just a number; it's an ordered pair that tells you both the input (x) and output (f(x)) at the function's peak. The maximum value here is 4 when x equals 2.

Piecewise Function Graphing

Piecewise functions are simply functions that have different expressions based on different intervals of the input value. Think of them as a combination of multiple mini-functions, each with its own rule.

For an absolute value function like f(x) = -2|x - 2| + 4, you can break it down into two linear functions—one for when 'x - 2' is positive and one for when 'x - 2' is negative. This results in a 'V' shaped graph with a sharp turn at the vertex. When you graph piecewise functions, carefully consider each piece and its domain. Then, merge the pieces to form a complete graph, ensuring continuity where the pieces meet.

Absolute Value Function Properties

Absolute value functions have distinct properties that set them apart. They create a 'V' shaped graph, which is always symmetrical about a vertical line through its vertex. The absolute value function represents distance, so it's never negative.

In f(x) = -2|x - 2| + 4, the 'V' is flipped upside down (due to the negative coefficient) and the function decreases in both directions from the vertex. Also, the graph of an absolute value function is made up of two linear pieces, making the slope of each piece on either side of the vertex a significant characteristic. In addition, these functions do not have a minimum or maximum provided they are not restricted to a domain or range, as they extend infinitely in at least one direction.