Problem 102 - Brief Applied Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Functions

The first function is \( f(x) \), which we'll consider as a general function. The second function is \( f(x+10)+10 \). Our goal is to compare these two functions and determine how their graphs are different.

Step 2: Translate the Function Horizontally

The expression \( f(x+10) \) implies a horizontal shift to the left by 10 units. In general, \( f(x+c) \) translates the graph of \( f(x) \) to the left by \( c \) units if \( c \) is positive.

Step 3: Translate the Function Vertically

The expression \( f(x+10)+10 \) adds 10 to the result of \( f(x+10) \), which translates the graph vertically upwards by 10 units. In general, adding \( d \) to \( f(x) \) shifts the graph vertically by \( d \) units.

Step 4: Combine the Transformations

Combining the two transformations from Steps 2 and 3, the graph of \( f(x+10)+10 \) is the graph of \( f(x) \) shifted 10 units to the left and 10 units up.

Key Concepts

Horizontal Shift

When comparing the two functions given, we can see that the function transformation involves a horizontal shift. In the function \( f(x + 10) \), the graph of the original function \( f(x) \) is shifted horizontally. This specifically means the graph will move to the left by 10 units.

Horizontal shifts result from changes made inside the function's argument, such as \( f(x + c) \). Here:

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

It's essential to note that this shift does not alter the graph's shape. It simply changes the position along the x-axis. Understanding this can significantly help when you sketch graphs of transformed functions.

Vertical Shift

Additionally, the function \( f(x+10) + 10 \) involves a vertical shift. This transformation occurs when a constant is added to the whole function, like the +10 in this case.

Vertical shifts can be described as follows:

• Adding a positive number \( d \) results in a shift upwards by \( d \) units.
• Adding a negative number would shift the graph downwards.

This transformation affects the y-values by increasing or decreasing all of them uniformly, making the graph move up or down along the y-axis. Importantly, this vertical shift, similar to the horizontal shift, does not change the shape of the graph but merely its position.

Graph Transformations

Graph transformations involve changing the position or shape of a graph on the coordinate plane. In the context of the functions \( f(x) \) and \( f(x+10)+10 \), two main types of transformations were applied: horizontal and vertical shifts.

The combined transformation results in:

• A horizontal shift to the left by 10 units, due to \( f(x+10) \).
• A vertical shift upwards by 10 units, due to the +10 added to \( f(x+10) \).

Together, these two shifts alter the graph's location, but keep its characteristic shape unchanged. Knowing how these transformations work allows you to understand and predict how the graph of a function will look after modifications. Graph transformations are vital tools in many areas, including physics, engineering, and economics.