Problem 52 - Intermediate Algebra

Question

For Problems \(35-52\), graph each exponential function. $$ f(x)=3^{-x+1} $$

Step-by-Step Solution

Answer

f(x) = 3^{-(x-1)} is a decreasing exponential function with rightward shift of 1 unit.

Step 1: Understand the Function

The given function is an exponential function in the form of \( f(x) = 3^{-x+1} \). We need to simplify this to better understand how it behaves. This function can be rewritten as \( f(x) = 3^{-(x-1)} \).

Step 2: Identify the Basic Exponential Form

The expression \( 3^{-(x-1)} \) indicates a horizontal shift of the basic function \( 3^{-x} \). The negative sign in the exponent indicates that it is a decreasing exponential function.

Step 3: Determine the Transformation

The expression \( x-1 \) tells us that the graph of \( 3^{-x} \) is shifted to the right by 1 unit. This means for every \( x \), the corresponding \( y \) value is that of the function \( 3^{-(x-1)} \).

Step 4: Plot Key Points

Select a few key points to easily graph the function. For example, when \( x = 1 \), \( f(x) = 3^{-(1-1)} = 3^0 = 1 \). When \( x = 2 \), \( f(x) = 3^{-(2-1)} = 3^{-1} = \frac{1}{3} \). Similarly, calculate other points, such as \( x = 0 \), \( x = -1 \).

Step 5: Sketch the Graph

Start plotting the points on a coordinate plane (using key points from Step 4). The basic graph will be decreasing from left to right, and passing through points such as (0, 3), (1, 1), and so on. Connect the points with a smooth curve.

Key Concepts

Graphing Exponential Functions

Exponential functions are a fascinating and important area of mathematics. These functions have the form \( f(x) = a^{x} \), where \( a \) is a positive real number. They are distinct because of how they steadily grow or shrink, dependent on the base \( a \). For exponential growth, \( a > 1 \), leading to the function increasing quickly as you move along the x-axis.
In this exercise, we look at \( f(x) = 3^{-x+1} \) that can be understood better by recognizing it as \( 3^{-(x-1)} \). Instead of just a plain multiplicative growth, it involves transformations, resulting in graphical shifts. Identifying and plotting key points is essential to correctly graph any exponential function. By evaluating the function for different x-values, such as \( x = 0, 1, 2 \), and plotting these points, you can sketch a curve showing how the function behaves across a range.
This is the core of graphing exponential functions: connecting calculated points smoothly and identifying the overall trend of increase or decrease depending on the function's direction.

Transformations of Functions

Transformations change the position or shape of a graph. For exponential functions, transformations often involve horizontal or vertical shifts, reflections, stretches, or compressions.
In the function \( f(x) = 3^{-(x-1)} \), the term \( x-1 \) indicates a horizontal shift. It's a right shift by 1 unit because you substitute \( x \) with \( (x - 1) \). Such shifts rearrange the graph along the x-axis without affecting its general exponential shape.
By understanding these transformations, you can easily anticipate the graph's appearance, even for more complex functions. This visualization and understanding of the shift are crucial for translating mathematical functions into understandable visuals.

Decreasing Exponential Graph

A decreasing exponential graph occurs when the base of the exponent is less than 1 in its effective term, such as when an exponent has a negative sign in \( f(x) = 3^{-x} \). It’s the mirror image in behavior of an increasing exponential graph, decreasing as you move from left to right along the x-axis.
For the function \( f(x) = 3^{-x+1} \), the negative exponent is key. It transforms our growth function into a decay function. Instead of rising, it falls, and this determines the downward slope observed when the graph is plotted.
Typically, a decreasing exponential graph begins high on the y-axis and approaches but never touches the x-axis as you move right. This continual approach to zero, without crossing, is one of the hallmark traits of exponential decay, reflecting how the quantities change repeatedly by the same proportion.