Problem 29 - Brief Applied Calculus

Question

$$ \text { Graph each function } $$ $$ f(x)=|x-3|-3 $$

Step-by-Step Solution

Answer

The function \( f(x) = |x-3| - 3 \) is a V-shaped absolute value graph with vertex at \( (3, -3) \).

Step 1: Understand the Function

The function given is \( f(x) = |x-3| - 3 \). This is an absolute value function, where \( |x-3| \) represents a V-shaped graph that is shifted according to the operations inside the absolute value and outside of it.

Step 2: Identify the Basic Absolute Value Function

The parent function for our given function is \( g(x) = |x| \). This graph is V-shaped, with its vertex at the origin, and is symmetrical about the y-axis.

Step 3: Determine the Horizontal Shift

The term \( |x-3| \) indicates a horizontal shift of 3 units to the right. This is because \( x-3 \) sets the inside expression to zero at \( x = 3 \), shifting the vertex of the graph from \( (0,0) \) to \( (3,0) \).

Step 4: Determine the Vertical Shift

The term \( -3 \) outside of the absolute value function indicates a vertical shift of 3 units downward. Thus, the entire graph is moved down, changing the vertex from \( (3,0) \) to \( (3,-3) \).

Step 5: Plot Key Points

With the vertex at \( (3,-3) \), calculate a few key points to ensure accuracy. For example, at \( x=1 \), \( f(x) = |1-3| -3 = 2 - 3 = -1 \); at \( x=3 \), \( f(x) = |3-3| -3 = -3 \); at \( x=5 \), \( f(x) = |5-3| -3 = 2 -3 = -1 \). Plotting these points helps visualize the V shape.

Step 6: Sketch the Graph

Using the vertex \( (3,-3) \) and key points \( (1,-1) \) and \( (5,-1) \), sketch the V-shaped graph. The graph opens upwards and is symmetrical about the vertical line \( x = 3 \). This completes the graphing of the function.

Key Concepts

Horizontal Shift

In the world of mathematics, a horizontal shift refers to moving a graph left or right on the coordinate plane. This concept is crucial when dealing with absolute value functions such as our example, \( f(x) = |x-3| - 3 \). Here, the expression within the absolute value, \( |x-3| \), contains the term \( x-3 \) which impacts the horizontal positioning of the graph.
Understanding horizontal shifts helps in adjusting the graph's location on the x-axis. For a term in the form of \( x-c \), the graph will shift \( c \) units to the right. In our function, \( x-3 \) indicates a shift of 3 units to the right. This means every point on the graph moves 3 units rightward from where it would be on the basic function \( g(x) = |x| \).

• Original position: Vertex at \( (0,0) \)
• Shift: Now vertex at \( (3,0) \)
• Reason: Solving \( x-3=0 \) gives \( x=3 \)

This horizontal shift effectively moves the entire V-shape of the absolute value function, helping visualize changes in its position.

Vertical Shift

Vertical shifts affect the height of the graph on the coordinate plane, either moving it up or down. This is easily recognized in our function \( f(x) = |x-3| - 3 \) by examining the term outside the absolute value bars. Here, the \(-3\) indicates a vertical shift downward by three units.
Such shifts are straightforward: when a constant is subtracted from or added to the function, it moves the graph without altering its shape. Specifically, a minus will move it down, while a plus moves it up.

• Initial vertex after horizontal shift: \( (3,0) \)
• Vertical shift: Down 3 units
• New vertex position: \( (3,-3) \)

Vertical shifts adjust where on the y-axis the main aspects of the graph sit, maintaining the graph's characteristics otherwise. This helps in understanding not just movement, but also graph positioning.

Vertex of the Graph

The vertex of an absolute value graph, like \( f(x) = |x-3| - 3 \), represents the point where the V-shape changes direction. It’s crucial for generating an accurate sketch as it marks the graph's lowest point when it opens upwards.
To find it, address both horizontal and vertical shifts. The parent function \( f(x) = |x| \) begins at the origin \((0,0)\). First, apply the horizontal shift, moving it to \((3,0)\). Then, account for the vertical shift by adjusting it to \((3,-3)\). These shifts combine to give the finalized vertex position.
Understanding the vertex helps in several ways:

• It shows the graph's central pivot point.
• The symmetry line can be drawn through the vertex: x = 3.
• Knowing its position allows proper graph scaling and translation.

The vertex thus offers a guideline for the graph's correct orientation and symmetry, providing a central anchor to other features.