Problem 3 - Algebra and Trigonometry Real Mathematics, Real People

Question

What type of transformation of the graph of \(f(x)=5^{x}\) is the graph of \(f(x+1) ?\)

Step-by-Step Solution

Answer

The graph of \(f(x+1)\) is a horizontal shift or translation of the graph of \(f(x) = 5^x\) one unit to the left.

Step 1: Recognize the Transformation

In the function \(f(x+1)\), the transformation is happening to the input or the 'x' value. That is, we are adding 1 to every x-value that we input into our function.

Step 2: Interpret the Transformation

In general, if we replace \(x\) with \(x+c\) in a function, the graph of the function shifts c units to the left. So for the function \(f(x+1)\), this means the graph of \(f(x) = 5^x\) will shift one unit to the left.

Step 3: Confirm the Transformation

Let's confirm this by testing a value. Take \(x=1\) for example, then \(f(1) = 5^1 = 5\), whereas \(f(1+1) = f(2) = 5^2 = 25\). When \(x\) was increased by 1, the resulting value became that of where \(x=2\) in the original function, effectively shifting the graph one unit to the left.

Key Concepts

Graph Shifts

When discussing the transformation of functions, one fundamental and commonly encountered modification is the horizontal shift of a graph. A graph shift occurs when every point of a function moves in the same direction along the x or y-axis. In the context of our function,

• The transformation of the graph of the function \( f(x) = 5^x \) to \( f(x+1) \) involves a horizontal shift.
• This particular shift is to the left.

This is because we add 1 to the input variable, \( x \), turning it into \( x + 1 \). A general rule to remember with horizontal shifts is:

• If you replace \( x \) with \( x + c \) in a function, the graph shifts \( c \) units to the left.
• If you replace \( x \) with \( x - c \), it shifts \( c \) units to the right.

Visualizing this can be particularly useful in understanding different transformations and effectively graphing functions.

Exponential Functions

Exponential functions are a crucial part of mathematics with profound implications in science, finance, and many other areas. An exponential function can be written in the form of \( f(x) = a^x \), where:

• \( a \) is a constant base greater than 0 and not equal to 1.
• \( x \) is the exponent, which is the variable that determines how many times the base is multiplied by itself.

The base determines the growth or decay rate:

• If \( a > 1 \), the function exhibits exponential growth.
• If \( 0 < a < 1 \), it shows exponential decay.

For our specific function, \( f(x) = 5^x \), with a base of 5, the function demonstrates rapid growth as \( x \) increases.Understanding these characteristics of exponential functions is essential when analyzing their graph and predicting behavior over time.

Function Transformation

Function transformation involves changing the appearance or position of a graph by altering its equation in specific ways.For instance, in our exercise with the function \( f(x) = 5^x \), transforming to \( f(x+1) \) changes how the function behaves:

• This change results from modifying the function's input by adding 1, effectively shifting the graph horizontally.
• The transformation doesn't change the scale or shape but influences the graph's position.

Well-known types of function transformations include:

• Shifting: Moving the graph horizontally or vertically.
• Stretching or Compressing: Changing the function's rate of increase or decrease.
• Reflecting: Flipping the graph across axes.

By mastering these transformations, one can easily tackle complex functions and sketch their graphs more effectively, a skill useful across various mathematical contexts.