Problem 32 - College Algebra with Corequisite Support

Question

Use a graphing utility to graph \(f(x)=10|x-2|\) on the viewing window [0,4] . Identify the corresponding range. Show the graph.

Step-by-Step Solution

Answer

The range of the function on [0, 4] is [0, 20].

Step 1: Understand the Function

The given function is \( f(x) = 10|x - 2| \). This is an absolute value function that is transformed compared to the basic \( y = |x| \). The function scales the output by 10 and shifts the graph 2 units to the right.

Step 2: Determine the Vertex

For the expression \( |x - 2| \), the vertex occurs at \( x = 2 \) where \( |x - 2| = 0 \). For \( f(x) = 10|x - 2| \), the vertex is at the point \( (2, 0) \).

Step 3: Evaluate the Function at Key Points

To understand the graph within the interval [0, 4], evaluate \( f(x) \) at \( x = 0 \), \( x = 2 \), and \( x = 4 \):- At \( x = 0 \), \( f(x) = 10|0 - 2| = 20 \).- At \( x = 2 \), \( f(x) = 10|2 - 2| = 0 \).- At \( x = 4 \), \( f(x) = 10|4 - 2| = 20 \).

Step 4: Sketch the Graph

Plot the points on the graph: \((0, 20), (2, 0), (4, 20)\). Connect these points with straight lines, forming a 'V' shape centered at \( (2, 0) \). The graph is symmetric about the line \( x = 2 \).

Step 5: Identify the Range

The graph's lowest point is at the vertex, \( y = 0 \), and it increases indefinitely upwards to \( y = 20 \) at both \( x = 0 \) and \( x = 4 \). The range of \( f(x) \) on the interval \([0, 4]\) is \([0, 20]\).

Step 6: Finalize the Graphing Solution

Using a graphing utility, plot the function and verify that it meets the manually determined coordinates and shape within the interval [0,4]. The graph aligns with the calculations, showing a V-shape with a vertex at (2, 0).

Key Concepts

Using a Graphing Utility

A graphing utility is a tool that helps visualize mathematical functions easily. It essentially plots the function for you, allowing a clearer understanding of how it behaves over a certain interval. To use a graphing utility effectively, you first need to input the function. In this case, it's the absolute value function: \(f(x) = 10|x - 2|\). Once the function is entered, set your viewing window. The viewing window determines the part of the function you'll see on the graph. Here, our window is set from \(x = 0\) to \(x = 4\). This will show you how the graph behaves between these points. Upon inputting this function into a graphing utility, you'll see a V-shaped graph centered at \((2, 0)\). This confirms the symmetry and shape that we analyzed manually. Using such utilities not only saves time but also significantly aids your understanding of function dynamics by providing visual representation.

Range Identification of a Function

Identifying the range of a function involves finding all possible output values (y-values) for the function. For our absolute value function \(f(x) = 10|x - 2|\), we're interested in outputs that it produces between \(x = 0\) and \(x = 4\). Let's examine these more closely.The key points calculated in the solved exercise are crucial for determining the range:

• At \(x = 0\), \(f(x) = 20\)
• At vertex \(x = 2\), \(f(x) = 0\)
• At \(x = 4\), \(f(x) = 20\)

These key evaluations show that the output starts at \(20\), drops to \(0\) at the vertex, then climbs back up to \(20\). Therefore, the range of \(f(x)\) in the specified domain is \([0, 20]\). Knowing the range helps to understand the behavior and limits of the function.

Function Transformation

Function transformation involves altering the basic shape or position of a graph through changes in the function itself. For \(f(x) = 10|x - 2|\), notice the two major transformations compared to the base absolute value function \(y = |x|\).Firstly, the function is horizontally shifted. The expression \(|x - 2|\) moves the graph 2 units to the right from the origin. This shift helps to place the vertex, the minimum point of the absolute value function, at \((2, 0)\).Secondly, there is a vertical scaling by a factor of 10. Instead of changing shape, this affects how steep or flat the graph appears. Here, multiplying \(|x - 2|\) by 10 stretches the V-shape vertically, making outputs grow larger per unit change in \(x\). These transformations highlight the flexibility in manipulating functions to represent various real-world scenarios by tweaking and adjusting their graphical representation.