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Chapter 1: Problem 540

For the following exercises, sketch a graph of the given function. $$ f(x)=4[|x-2|-6] $$

Short Answer

Expert verified
The graph of the function \(f(x)=4[|x-2|-6]\) is a V-shaped graph that opens upwards, has a vertex at (2, -24), and stretches vertically by a factor of 4 and shifts downward by 24 units.

Step by step solution

01

Identify the Type of Function

The given function is \( f(x) = 4[|x - 2| - 6] \), which involves an absolute value function. This type of function will create a V-shaped graph. The expression inside the absolute value gives a horizontal shift, while the expression outside affects vertical shifts and scaling.
02

Determine Key Transformations

The function \( f(x) = 4[|x - 2| - 6] \) can be broken down as follows:1. \(|x - 2|\) shifts the graph 2 units to the right.2. Subtracting 6 (\

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graph Sketching
Graph sketching is a valuable skill that allows us to understand the behavior of functions visually. For the function given, \( f(x) = 4[|x - 2| - 6] \), it is crucial to first recognize that it is based on the absolute value function, \( |x| \). The absolute value function typically forms a V-shaped graph centered at the origin. In this case, transformations will change its position and shape:
  • Horizontal and vertical shifts will move the graph from its standard position.
  • The coefficient 4 in front of the expression alters the steepness of the graph.
When sketching, start by finding key points, such as where the absolute value expression equals zero. This helps find the vertex, the turning point of the V-shape. Then, plot additional points on either side of the vertex to reflect the effect of the scaling factor (4 here). With these points, you can accurately trace the graph's path on the coordinate plane.
Horizontal Shift
Understanding the horizontal shift is a core part of transforming functions. The expression \( |x - 2| \) indicates that the absolute value graph shifts horizontally. Normally, \( |x| \) is centered at the y-axis, but \( |x - 2| \) results in moving the graph 2 units to the right. This shift occurs because the change affects the input before the absolute value is applied.To visualize this:
  • Consider the standard V-shaped graph of \( |x| \) at the origin.
  • Shift every point on this graph exactly 2 units to the right, maintaining the same vertical positions.
As a result, the graph鈥檚 vertex, which was originally at the origin, moves to the point \((2, 0)\). Such transformations are crucial for accurately sketching modified functions and understanding how they interact with the x-axis.
Vertical Shift
A vertical shift involves moving the entire graph up or down along the y-axis. In the function \( f(x) = 4[|x - 2| - 6] \), there is a subtraction of 6 within the brackets, indicating a vertical shift. Specifically, this means the entire graph is moved 6 units downwards.For a step-by-step breakdown:
  • Begin with the horizontally shifted graph from \( |x - 2| \) which has its vertex at \((2, 0)\).
  • Now, "subtract" 6 from every y-value on this graph to shift it down.
  • This shifts the vertex from \((2, 0)\) to \((2, -6)\).
Vertical shifts do not alter the shape of the graph, only its vertical position. Incorporating both the horizontal and vertical shifts, you can see how the transformations work together to position the graph precisely in the coordinate system. This understanding is essential for handling and interpreting modified absolute value functions.

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Most popular questions from this chapter

For the following exercises, use each pair of functions to find \(f(g(0))\) and \(g(f(0))\) $$ f(x)=\sqrt{x+4}, g(x)=12-x^{3} $$

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Given \(f(x)=2 x^{2}+1\) and \(g(x)=3 x-5,\) find the following: \(\begin{array}{ll}{\text { a. }} & {f(g(2))} \\ {\text { b. }} & {f(g(x))} \\\ {\text { c. }} & {g(f(x))} \\ {\text { d. }} & {(g \circ g)(x)} \\\ {\text { e. }} & {(f \circ f)(-2)}\end{array}\)

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Given \(f(x)=x^{2}+2 x\) and \(g(x)=6-x^{2},\) find \(f+g, f-g, f g,\) and \(\frac{f}{g}\) . Determine the domain for each function in interval notation.
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