Problem 10 - College Algebra

Question

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 of \(y=f(x-49)\) is the graph of \(f(x)\) shifted 49 units to the right.

Step 1: Understand the Problem

We have a function transformation problem where we need to describe how the function \(y = f(x-49)\) transforms the original function \(f(x)\).

Step 2: Identify the Type of Transformation

The transformation in the given function \(y = f(x-49)\) involves a horizontal shift. Identifying this as a horizontal translation based on the subtraction inside the function's argument is crucial.

Step 3: Determine the Direction of the Shift

In the function \(y = f(x-49)\), the \(-49\) indicates a horizontal shift. When the function \(f\) is transformed to \(f(x-a)\), it results in a shift to the right by \(a\) units. Thus, here, \(f\) moves 49 units to the right.

Key Concepts

Horizontal Shift

A horizontal shift is a specific type of graph transformation that involves moving the graph of a function left or right on the coordinate plane. In the case of the function transformation from \(f(x)\) to \(f(x-49)\), we are dealing with a horizontal shift to the right.
When we see a function in the form \(f(x-a)\), the shift is always to the right by \(a\) units if \(a\) is positive.
Conversely, if the function appears as \(f(x+a)\), the graph shifts to the left by \(a\) units.

• Positive shift: Right by "a" units.
• Negative shift: Left by "a" units.

For \(f(x-49)\), we subtract 49 inside the function, indicating the graph of \(f(x)\) is shifted 49 units to the right.

Graph Transformation

Graph transformation involves altering the appearance of the graph of a function. This can be done through stretching, compressing, reflecting, or shifting. A horizontal transformation, like the one seen in \(y=f(x-49)\), specifically changes the position of the graph along the x-axis.
Understanding graph transformations is important for visual analysis and function manipulation.

• Vertical transformations adjust the height or orientation.
• Horizontal transformations adjust the position horizontally.
• Combinations of these transformations can occur.
• Vertical and horizontal shifts change the location but not the shape of the graph.

Translation

Translation refers to the process of moving every point of a figure or a graph a constant distance in a specified direction. In the context of function transformation, translation involves moving the graph horizontally or vertically.
When translating a function graph horizontally, like in \(y=f(x-49)\), every point on the graph is shifted to the right by 49 units.
Translation differs from other transformations as it maintains the shape and orientation of the graph. The only change is the graph's position.

• Horizontal translation: Shift graph left or right.
• Vertical translation: Shift graph up or down.

Function Translation

Function translation is a type of transformation where the entire graph of a function moves to a new position. Specifically, in \(y=f(x-49)\), this involves a horizontal translation where the graph is shifted without any change to its shape.
This concept is crucial for understanding how functions behave under transformations.

• Maintains overall graph shape.
• Does not affect the graph's orientation or scale.

While the focus here is on horizontal translation, it's important to note that translation can occur in any direction on a plane. This is part of what makes function transformations powerful tools in mathematical analysis.