Problem 41 - Precalculus

Question

Graph the exponential function using transformations. State the \(y\) -intercept, two additional points, the domain, the range, and the horizontal asymptote. $$f(x)=e^{x+1}-4$$

Step-by-Step Solution

Answer

The y-intercept is (0, -1.282), with additional points (1, 3.389) and (-1, -3). The domain is all real numbers, the range is (-4, ∞), and the horizontal asymptote is y = -4.

Step 1: Identify the Base Function

The base function for transformations is the exponential function \(f(x) = e^x\). The graph of \(f(x) = e^x\) passes through the point \((0, 1)\) and has a horizontal asymptote at \(y = 0\).

Step 2: Apply Horizontal Transformation

The expression \(e^{x+1}\) represents a horizontal shift to the left by 1 unit of the base function \(e^x\). To verify, note that if \( f(x) = e^x\), then \(f(x+1) = e^{x+1}\) will shift the graph leftwards.

Step 3: Apply Vertical Transformation

The subtraction of 4, i.e., \(-4\) in \(f(x) = e^{x+1} - 4\), is a vertical shift downwards by 4 units. Thus, the entire graph of \(e^{x+1}\) is lowered by 4 units.

Step 4: State the Transformed y-intercept

For the y-intercept, set \(x = 0\) and compute: \[ f(0) = e^{0+1} - 4 = e^1 - 4 \approx 2.718 - 4 = -1.282.\]Thus, the y-intercept is \((0, -1.282)\).

Step 5: Determine Two Additional Points

Choose two other values of \(x\) to find additional points:1. For \(x=1\), \[ f(1) = e^{1+1} - 4 = e^2 - 4 \approx 7.389 - 4 = 3.389. \] Thus, the point is \((1, 3.389)\).2. For \(x=-1\), \[ f(-1) = e^{-1+1} - 4 = e^0 - 4 = 1 - 4 = -3. \] Thus, the point is \((-1, -3)\).

Step 6: State the Domain and Range

The domain of an exponential function is all real numbers, or \((-f, f )\). The vertical shift affects the range, making it \((-4, f)\).

Step 7: Identify the Horizontal Asymptote

The horizontal shift does not affect the asymptote. However, the vertical shift downward by 4 units moves the asymptote from \(y = 0\) to \(y = -4\).

Key Concepts

Graphing Transformations

Graphing transformations involves shifting, stretching, or reflecting a graph compared to its base function. Here, we work with the exponential function, which has unique properties that make transformations straightforward yet important.

• Horizontal shifts move the graph left or right.
• Vertical shifts move the graph up or down.
• Reflections flip the graph over a specific axis.

In the exercise, the base exponential function is \[ f(x) = e^x \]The transformations applied are:- **Horizontal Shift**: The function is shifted left by 1 unit, due to the \(x + 1\) in the exponent.- **Vertical Shift**: The entire function moves down 4 units, due to the -4 at the end. These changes significantly alter the original graph while maintaining the general exponential growth pattern.

Domain and Range

Understanding the domain and range of a function helps us comprehend where the function is defined and the values it can take.

• The **domain** of exponential functions like \( f(x) = e^x \) is all real numbers, symbolized by \( (-\infty, \infty) \).
• The **range** is transformed by vertical shifts. For our function, the transformation shifted the graph downward. As a result, the range changes from \( (0, \infty) \) to \( (-4, \infty) \).

These parameters ensure that while any real number can be inputted, the output will always be greater than -4 due to the vertical shift.

Horizontal Asymptote

An exponential function generally approaches a horizontal line, known as an asymptote, as it progresses towards extremes on the x-axis.

• For the base function \( f(x) = e^x \), the horizontal asymptote is at \( y = 0 \).
• The vertical transformation affects the horizontal asymptote.For our function, it shifts down 4 units, from \( y = 0 \) to \( y = -4 \).

This means as \( x \) moves toward negative infinity, the graph approaches but never actually reaches \( y = -4 \). The concept of a horizontal asymptote is crucial because it tells us about the ultimate behavior of the graph, providing insights even when the precise value seem distant.

Exponential Functions

Exponential functions, such as \( f(x) = e^x \), describe rapid growth or decay processes, frequently occurring in real-world applications.Key features of exponential functions include:

• **Base Value**: The constant \( e \) is approximately 2.718, a natural exponential base widely used in mathematics and other scientific applications.
• **Growth Pattern**: The graph of an exponential function steadily increases (for positive exponents) or decreases (for negative exponents).
• **Transformations**: They can undergo various transformations which alter their graph, including shifting and reflecting. This transforms the behavior while preserving the primary exponential nature.

Understanding these aspects allows you to apply exponential functions to model various phenomena, from population dynamics to radioactive decay. By leveraging graphing transformations, these functions can elegantly express complex situations in a digestible form.