Problem 50 - Intermediate Algebra

Question

For Problems \(35-52\), graph each exponential function. $$ f(x)=2^{x}+1 $$

Step-by-Step Solution

Answer

Plot key points such as (0,2) and (1,3), and draw a curve approaching the asymptote at y=1.

Step 1: Understand the Function

The given function is \( f(x) = 2^x + 1 \). This is an exponential function where the base is 2 and there is a vertical shift upward by 1 unit. Understanding this helps in predicting the shape and position of the graph.

Step 2: Determine Key Characteristics

Identify the y-intercept and other key points. For \( x = 0 \), \( f(0) = 2^0 + 1 = 1 + 1 = 2 \). Since \( 2^x \) never becomes negative, the minimum value of \( f(x) \) is 1 when the graph approaches \( x = 0 \).

Step 3: Calculate Additional Points

Choose additional \( x \)-values to calculate corresponding \( y \)-values. For example, \( f(1) = 2^1 + 1 = 3 \), \( f(2) = 2^2 + 1 = 5 \), and \( f(-1) = 2^{-1} + 1 = \frac{1}{2} + 1 = 1.5 \).

Step 4: Identify Asymptote

The equation \( f(x) = 2^x + 1 \) reveals a horizontal asymptote at \( y = 1 \) because as \( x \to -\infty \), \( 2^x \to 0 \), so \( f(x) \to 1 \).

Step 5: Graph the Function

Using a graphing tool, plot the points determined in the previous steps: (0, 2), (1, 3), (2, 5), and (-1, 1.5). Draw a smooth curve through these points that approaches the horizontal asymptote \( y = 1 \) as \( x \to -\infty \) and increases rapidly as \( x \to \infty \).

Key Concepts

Graphing Exponential Functions

Graphing an exponential function involves understanding its unique characteristics. Exponential functions have the general form \( f(x) = a^x \) where \( a \) is a positive constant. They exhibit rapid growth or decay depending on the base. This rapid change is what makes them so distinct and useful in modeling real-world phenomena like population growth or radioactive decay.
To graph these functions:

• Identify the base \( a \). A base greater than 1 indicates exponential growth, while a base between 0 and 1 indicates decay.
• Find key points, such as the y-intercept at \( x = 0 \).
• Determine several additional points to establish the function’s trend.
• Draw a smooth curve through the plotted points.

Remember, the curve of an exponential function is symmetric around its respective axis if no horizontal or vertical shifts are applied. This concept helps in sketching the graph precisely and identifying behavior of the function over different values of \( x \).

Horizontal Asymptotes

Horizontal asymptotes in exponential functions play a significant role in understanding the behavior of the graph as \( x \) approaches extreme values. For an exponential function of the form \( f(x) = a^x + c \), the horizontal asymptote is located at \( y = c \).
This line represents a boundary the function's graph approaches but never actually touches as \( x \) tends towards negative infinity. For example, in the function \( f(x) = 2^x + 1 \), the graph approaches the line \( y = 1 \) but never crosses it.
Key aspects of horizontal asymptotes:

• They provide a limit beyond which the function will not go as \( x \to -\infty \).
• They help in visualizing the end-behavior of the graph, aiding in sketches and predictions.

Recognizing horizontal asymptotes can provide valuable insights into the long-term behavior of the function, which is crucial when modeling scenarios that extend over long periods or substantial ranges of \( x \).

Vertical Shifts

Vertical shifts occur when a constant is added or subtracted from an exponential function. This merely shifts the graph up or down on the y-axis. In the case of \( f(x) = 2^x + 1 \), the graph is shifted upwards by 1 unit.
The effects of vertical shifts:

• The entire graph moves without changing its shape, slope, or horizontal asymptote.
• The y-intercept changes accordingly; in this example, it moves from (0, 1) to (0, 2).

Vertical shifts can make a significant difference in interpreting the function's behavior in context, especially in situations where an initial value or baseline is crucial for analysis. They allow us to adjust the function's baseline level, providing flexibility in how we describe real-world phenomena.

Function Transformations

Function transformations involve various alterations that modify the appearance or position of a graph. For exponential functions, these transformations can include horizontal and vertical shifts, reflections, and stretches or compressions.
Common transformations:

• Reflections: These occur across the x-axis or y-axis, changing the direction of the graph’s growth or decay.
• Stretching/Compressing: Modifying the coefficient in front of the base (e.g., \( f(x) = a \cdot 2^x \)) stretches or compresses the graph vertically, affecting its steepness.

Understanding these transformations can enhance your ability to manipulate and generalize the behavior of exponential functions. By mastering them, you can confidently adjust any graph to fit specific conditions or interpret changes in data over different scenarios.