Problem 10 - College Algebra

Question

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function \(f\). $$ y=f(x-49) $$

Step-by-Step Solution

Answer

The graph is shifted 49 units to the right.

Step 1: Identify the Original Function

The original function is given as \(y = f(x)\). This represents the base graph that we will be transforming.

Step 2: Recognize the Transformation

The transformation applied to the original function \(f(x)\) is in the form of \(y = f(x - c)\). In this case, \(c = 49\), which means that the graph of \(f(x)\) is shifted.

Step 3: Determine the Type of Transformation

Since the transformation is \(f(x - 49)\), this represents a horizontal shift. Specifically, the graph of the function \(f(x)\) shifts 49 units to the right. This is because the value inside the function's argument decreases as \(x\) increases by 49.

Step 4: State the Transformation

The graph of the function \(y = f(x-49)\) is a horizontal translation of the original graph \(y = f(x)\). The transformation results in moving the entire graph 49 units to the right along the x-axis.

Key Concepts

Horizontal Shift

A horizontal shift involves moving a graph left or right along the x-axis. When you encounter a function like \(y = f(x - c)\), it indicates a horizontal shift. Here, \(c\) represents how far the graph shifts.

If \(c\) is positive, the shift is to the right. For example, in \(y = f(x - 49)\), the graph moves 49 units to the right. If \(c\) was negative, you would shift the graph to the left instead.

• The value of \(c\) directly tells you the distance and direction of the shift.
• Think of "minus" inside the function as moving right, since you need a larger \(x\) to achieve the same output as before the shift.
• Conversely, a "plus" would result in moving left on the graph.

Understanding horizontal shifts is crucial because they show how changes inside the parentheses alter the graph's position. This sets the foundation for exploring and predicting shifts in more complex functions.

Function Transformation

Function transformation is all about changing a graph's position or shape. It's like giving your function a makeover! Transformations can include shifts, stretches, or reflections. These changes help us understand the function's behavior better.

Horizontal shifts are one type, moving graphs side to side without altering their shape. These are crucial when analyzing how functions react to changes in their variables. Other transformations involve adjusting inputs or outputs:

• Vertical shifts move graphs up or down by adding or subtracting from \(f(x)\).
• Stretches expand or compress the graph by multiplying or dividing \(f(x)\) or \(x\).
• Reflections flip the graph across an axis.

Mastering these transformations is helpful in predicting and understanding changes in data and real-world situations. They equip you to modify and interpret functions in diverse contexts.

X-Axis Translation

X-axis translation is essentially another term for a horizontal shift. It refers to moving a graph side to side, parallel to the x-axis. Translating along the x-axis can affect how the function aligns with data or other graphs.

With a function like \(y = f(x - 49)\), the graph translates 49 units right along the x-axis. This means every point on \(f(x)\) shifts rightward, aligning them forward without changing the graph's appearance.

• This translation leaves the y-axis intercept and overall structure of the graph unchanged.
• Translation helps maintain the original features of the graph while changing its position.

Understanding x-axis translation is useful as it helps in solving equations graphically and in creating models that fit specific scenarios. It refines the positioning of graphs for better data representation.