Problem 101 - Brief Applied Calculus

Question

$$ \text { How do the graphs of } f(x) \text { and } f(x+10) \text { differ? } $$

Step-by-Step Solution

Answer

The graph of \( f(x+10) \) is shifted 10 units left compared to \( f(x) \).

Step 1: Identify the Functions

The functions given are \( f(x) \) and \( f(x+10) \). They represent a standard function and its horizontal shift, respectively.

Step 2: Understand Horizontal Shifts

When you have a function \( f(x) \) and you add a constant inside the argument, like in \( f(x+c) \), the graph will shift horizontally. The rule is that \( f(x+c) \) will move the graph \( c \) units to the left if \( c > 0 \).

Step 3: Apply the Horizontal Shift to f(x+10)

The function \( f(x+10) \) is \( f(x) \) shifted by 10 units to the left. Adding 10 inside the function affects only the x-coordinate by moving each point on the graph of \( f(x) \) 10 units to the left.

Step 4: Conclusion by Comparing Graphs

The graph of \( f(x+10) \) is identical in shape and size to \( f(x) \), but every point on the graph of \( f(x) \) has been moved 10 units to the left in the graph of \( f(x+10) \).

Key Concepts

Horizontal Shifts

One of the basic transformations of functions is the horizontal shift. This transformation involves moving the graph of a function left or right along the x-axis, without changing its shape. When you see a function written as \( f(x + c) \), it indicates a horizontal shift. Here, \( c \) is the number of units the graph will shift.

• If \( c \) is positive, the graph shifts to the left by \( c \) units.
• If \( c \) is negative, the graph shifts to the right by \(-c\) units.

For example, with the function \( f(x+10) \), the entire graph of \( f(x) \) moves 10 units to the left. Although the position changes, the overall shape and size of the graph remain the same.

Graph Comparison

When comparing graphs of functions, look beyond the shapes and examine the transformations applied. The primary transformation in our exercise is the horizontal shift. Despite the horizontal movement, the graphs of \( f(x) \) and \( f(x+10) \) are identical in shape.
This means:

• Both graphs have the same height and width at corresponding points.
• Each point on \( f(x+10) \) is exactly 10 units left of its corresponding point on \( f(x) \).

Understanding graph comparison helps in visualizing the effect of transformations and in predicting how changes to functions will affect their graphs.

Function Notation

Function notation is a convenient way to express mathematical relationships and transformations. By using a function notation like \( f(x) \), we establish a clear rule that explains how to arrive at outputs from the given inputs, x-values.
Here are some key points about function notation:

• \( f(x) \) represents the original function, where each x-value is plugged into the function rule to obtain a corresponding \( y \)-value.
• Functions can be transformed by altering the inputs, as seen with \( f(x+c) \), resulting in predictable shifts in the graph.
• Clear function notation makes it easier to understand and predict the effect of adjustments, such as shifts or stretches, on the graph.

Using function notation not only clarifies transformations applied to the function but also simplifies communication and understanding in mathematics.