Problem 12 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, sketch the graph of the function. $$ f(x)=e^{2 x} $$

Step-by-Step Solution

Answer

The function \(f(x) = e^{2x}\) being an exponential function has a specific shape. It passes the point (0, 1), inclines as \(x\) increases, and the \(y\)-axis is its horizontal asymptote. This can be verified by calculating the function's values at various points and plotting them.

Step 1: Understanding the exponential function

Exponential functions of the form \(f(x) = a^x\), where \(a\) is a positive real number not equal to 1, have a characteristic shape. They always pass through the point (0, 1) and, in the case where \(a > 1\) , they increase, becoming steeper as \(x\) increases.

Step 2: Calculate the function's value at various points

Calculate the function's value at a few key points for sketching the graph. Some relevant points would be \(x = -2, -1, 0, 1, 2\). Then we compute the respective \(f(x)\) for those values: \(f(-2) = e^{-4}\), \(f(-1) = e^{-2}\), \(f(0) = e^0 = 1\), \(f(1) = e^2\), \(f(2) = e^4\).

Step 3: Plotting the function

Now we plot these calculated points along with the defining characteristics of the exponential function to accurately draw the function. The function should pass through the points (0,1), inclines as \(x\) increases and have an asymptote at \(y=0\)

Key Concepts

Graphing Functions

When graphing exponential functions like \( f(x) = e^{2x} \), it's important to understand their typical characteristics. Graphing involves plotting the function on a coordinate plane to visually represent its behavior. In the case of our function, it has an exponential form, meaning it rises sharply with increasing \( x \). This sharp rise is a hallmark of exponential growth.Graphs of exponential functions usually start below the x-axis for negative \( x \), cross the y-axis at some value, and continue upwards as \( x \) increases. These functions also have a horizontal asymptote, which is an invisible line that the function gets closer to but never actually touches, typically along the x-axis (or \( y = 0 \)).To sketch this function, you should:

• Identify key points: For \( f(x) = e^{2x} \), the graph passes through (0, 1), since \( e^0 = 1 \).
• Find additional points by calculating \( f(x) \) for several \( x \) values such as \( -2, -1, 0, 1, 2 \).
• Draw the graph starting from the left, moving upwards sharply, and approaching the horizontal asymptote without touching it.

Function Behavior

The behavior of an exponential function is defined by how it changes as the input \( x \) changes. For \( f(x) = e^{2x} \), this change is exponential, meaning its rate of increase grows faster as \( x \) becomes larger.Some key behavioral aspects of exponential functions include:

• **Monotonically increasing:** This function always increases because the base \( e \) (approximately 2.718) is greater than 1. As \( x \) increases, \( f(x) \) will never decrease.
• **Smooth curve:** The graph is continuous and smooth, with no breaks, jumps, or abrupt changes.
• **Y-intercept:** The function crosses the y-axis at \( y=1 \) because \( f(0) = e^0 = 1 \).

Understanding these characteristics can help in predicting how the graph will look and how the function behaves as \( x \) approaches positive or negative infinity.

Exponential Growth

Exponential functions like \( f(x) = e^{2x} \) represent exponential growth, a powerful concept used in various scientific fields and real-life scenarios. This growth is characterized by the quantity doubling quickly over equal intervals of time.In our function:

• \( e^{2x} \) indicates very rapid growth since the exponent is positive and multi-fold (twice the regular \( e^x \)). It grows even faster compared to simpler exponential functions like \( e^x \).
• This rapid growth is why exponential functions are frequently used to model populations, financial predictions, and natural processes, where quantities grow at increasing rates.

To better understand exponential growth, think about scenarios where small initial stages might seem negligible, but yielding significant consequences over time – just like the function \( f(x) = e^{2x} \) skyrocketing as \( x \) increases.