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Chapter 0: Problem 18

Sketch the graph of the equation by translating, reflecting, compressing, and stretching the graph of \(y=x^{2}, y=\sqrt{x}\), \(y=1 / x, y=|x|,\) or \(y=\sqrt[3]{x}\) appropriately. Then use a graphing utility to confirm that your sketch is correct. $$ y=1-|x-3| $$

Short Answer

Expert verified
The graph is a V-shape, opening downward, with the vertex at (3, 1).

Step by step solution

01

- Identify the Base Function

Recognize that the base function for the equation \( y = 1 - |x - 3| \) is \( y = |x| \). The task is to determine how \( |x| \) is transformed to obtain the given equation.
02

- Apply Horizontal Shift

The equation is \( y = 1 - |x - 3| \). Notice that inside the absolute value, we have \( x - 3 \). This represents a horizontal shift of the function \( y = |x| \) to the right by 3 units.
03

- Reflect Vertically and Shift Upward

After the horizontal shift, the equation becomes \( y = |x - 3| \). The next step is to apply the transformation outside the absolute value bars, which is \( 1 - |x - 3| \). This is equivalent to reflecting \( |x - 3| \) vertically over the x-axis (multiply by -1) and then shifting it upward by 1 unit.
04

- Confirm Transformations with the Graph Utility

Plot the final transformed equation \( y = 1 - |x - 3| \) using a graphing utility. Verify the transformations: right 3 units, reflected over the x-axis, and shifted up by 1 unit match your sketch. Check if the graph is V-shaped, opening downward, with the vertex at (3, 1).

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Key Concepts

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

Horizontal Shift
In graph transformations, a horizontal shift moves the entire graph to the left or right. This shift is determined by changes within the function's input. For example, in the equation \( y = 1 - |x - 3| \), the absolute value function \(|x|\) is shifted. Here, \( x - 3 \) suggests a horizontal shift to the right by 3 units. This is because you subtract 3 from \( x \), effectively moving all points on the graph 3 units to the right. Remember:
  • \( x - c \) shifts the graph right by \( c \) units.
  • \( x + c \) shifts the graph left by \( c \) units.
Understanding horizontal shift is crucial for graph transformation, as it helps you accurately position the graph on the coordinate plane.
Vertical Reflection
A vertical reflection involves flipping the graph of a function over the x-axis. This transformation is achieved by multiplying the entire function by -1. In our transformation of \( y = 1 - |x - 3| \), after shifting horizontally, we apply a vertical reflection. The term \(-|x - 3|\) reflects the graph of \(|x - 3|\) over the x-axis.

Here's how you recognize a vertical reflection:
  • If you see \(-f(x)\), the graph of \(f(x)\) is reflected over the x-axis.
  • The graph maintains its shape but is inverted vertically.
In this specific example, the V-shaped graph that typically opens upwards is reflected to open downwards.
Absolute Value Function
The absolute value function \( y = |x| \) forms a V-shaped graph with the vertex at the origin (0,0). This function is characterized by:
  • Non-negative outputs.
  • A sharp turn at the vertex.
The absolute value function acts as the base for many transformations. For instance, in \( y = 1 - |x - 3| \), the transformation is centered around \(|x|\). By shifting, reflecting, and shifting again, we derive the new graph from this base. Understanding \(|x|\) transformations helps you predict how the graph will shift, stretch, compress, or reflect when additional operations are applied.
Vertical Shift
A vertical shift moves the graph up or down on the coordinate plane. In our equation \( y = 1 - |x - 3| \), the constant \( +1 \) outside the absolute value function indicates a vertical shift:
  • A positive constant shifts the graph upward.
  • A negative constant shifts the graph downward.
After shifting \(|x - 3|\) horizontally and reflecting, the graph \(-|x - 3|\) is finally moved upward by 1 unit. This causes the vertex to rise from its original position. Vertical shifts are simple but essential in identifying the final position of the transformed graph on the y-axis. In this case, the graph's vertex ends up at (3, 1).

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