Problem 28 - Algebra and Trigonometry

Question

\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ f(x)=|x-3| $$

Step-by-Step Solution

Answer

The graph is a V-shape shifted 3 units right with vertex at (3,0).

Step 1: Identify the Standard Function

The given function is \( f(x) = |x-3| \). The standard form of an absolute value function is \( g(x) = |x| \). We start with this basic absolute value function which has a V-shape that opens upwards with its vertex at the origin (0,0).

Step 2: Apply Horizontal Transformation

The function \( f(x) = |x-3| \) represents a horizontal shift of the function \( g(x) = |x| \). To reflect this transformation, we move the graph of \( g(x) = |x| \) to the right by 3 units. This changes the vertex of the V-shape from (0,0) to (3,0).

Step 3: Graph the Transformed Function

Now, using the transformations, we can sketch the graph of \( f(x) = |x-3| \). Start with the vertex at the new location (3,0). Since the absolute value function is symmetric, the arms of the V extend outwards with a slope of 1 and -1 from the vertex.

Key Concepts

Graph Transformations

Graph transformations are essential techniques that allow us to modify the appearance of a given function's graph. This can help us understand how different functions relate to each other. Essentially, a transformation involves altering the graph of a function based on operations such as shifting, reflecting, stretching, or compressing.

• **Shifting:** Moves the graph horizontally or vertically without changing its shape.
• **Reflecting:** Flips the graph across an axis.
• **Stretching/Compressing:** Alters the distance of the graph from an axis either by stretching or squeezing.

To transform the graph of a function, it’s helpful to start with a basic form and apply these modifications. By recognizing the initial or standard function and using algebraic manipulations, you can determine how the transformation affects the graph. This skill is invaluable for sketching graphs quickly and accurately.

Horizontal Shifts

Horizontal shifts are a type of graph transformation that moves the graph left or right along the x-axis. This happens when a constant is added or subtracted from the input variable, typically represented as adjustments within the function's argument.
For example, in the function given as \( f(x) = |x-3| \), the \'-3\' indicates a horizontal shift to the right by 3 units. Here are the general rules:

• If you subtract a value from \(x\) in \(f(x) = |x - a|\), the graph shifts to the right by \(a\) units.
• If you add a value to \(x\) in \(f(x) = |x + a|\), the graph shifts to the left by \(a\) units.

This transformation does not affect the shape of the graph. For an absolute value function like \( |x| \), shifting changes only the position of the vertex, not the opening direction or slope of the line segments extending from the vertex.

Basic Graph Shapes

Understanding basic graph shapes is crucial for identifying functions and predicting their behavior. The absolute value function, denoted as \( g(x) = |x| \), serves as an excellent example.

• The graph of \( g(x) = |x| \) resembles a 'V' shape that opens upwards.
• Its vertex, or the point where the direction changes, is initially at the origin \( (0,0) \).
• Each arm of the 'V' has a slope of 1 (ascending line segment) and -1 (descending line segment).

The vertex is the key feature in understanding absolute value graphs because it indicates where the transformation has been applied. Recognizing the basic graph shape allows easy application of transformations to sketch new functions quickly by relocating the vertex as needed and extending the lines accordingly.