Problem 33 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the function.\(f(x)=e^{2 x}\)

Step-by-Step Solution

Answer

The graph of the function \(f(x) = e^{2x}\) is an upward curve in the first quadrant, passing through the point (0,1), and it approaches, but never reaches, the x-axis as \(x\) decreases.

Step 1: Understand the Function

The given function \(f(x)=e^{2x}\) is an exponential function. We know that \(e^{2x}\) grows faster than \(e^{x}\) for positive values of \(x\) and decreases to zero much quicker for negative values of \(x\). This should be reflected in our graph.

Step 2: Identify Key Points

Firstly, for \(x=0\), we have \(f(0)=e^{2*0}=1\), the y-intercept. Then, choose some values of \(x\) where \(x>0\), for instance, \(x=1\), and \(x=2\), and we have \(f(1)=e^{2}=7.39\), and \(f(2)=e^{4}=54.60\) respectively. For \(x<0\), choose \(x=-1\), and \(x=-2\), then we get \(f(-1)= e^{-2}=0.14\) and \(f(-2)=e^{-4}=0.018\), respectively.

Step 3: Sketch the Graph

Draw the Cartesian coordinate system and mark the key points calculated in the previous step. Then curve upwards in the positive direction along these plotted points, which reflects the nature of our exponential function \(f(x)=e^{2x}\). For \(x<0\), the curve should approach the x-axis but never touch or cross it, this is the asymptote of the function.

Key Concepts

Graph Sketching

Graph sketching is all about capturing the essence of a function visually. When dealing with exponential functions like \(f(x) = e^{2x}\), it's important to observe the rapid growth for positive values of \(x\) and swift approach to zero for negative values.
Start by drawing a set of axes. The horizontal axis is your \(x\)-axis, and the vertical one is the \(y\)-axis. Choose appropriate scales based on the values you have.
With exponential functions, you'll quickly notice steep growth.

• For positive \(x\), the function moves upwards rapidly.
• For negative \(x\), the line approaches closer to but never touches the \(x\)-axis.

To sketch, first plot the key points you identified, then connect them smoothly. The curve should ascend steeply in positive \(x\), creating a visual of exponential growth, and level out nearing the \(x\)-axis for negative \(x\). Remember, the graph will never cut across the \(x\)-axis due to the nature of exponential growth.

Key Points Identification

Identifying key points is crucial when sketching graphs. It allows you to capture pivotal values that define your graph's shape. For the function \(f(x) = e^{2x}\), start with the y-intercept which occurs at \(x = 0\).
At this point, \(f(0) = e^0 = 1\) — a critical anchor on your graph.

• For \(x = 1\), calculate \(f(1) = e^2 \approx 7.39\).
• For \(x = 2\), calculate \(f(2) = e^4 \approx 54.60\).
• For \(x = -1\), calculate \(f(-1) = e^{-2} \approx 0.14\).
• For \(x = -2\), calculate \(f(-2) = e^{-4} \approx 0.018\).

These points reflect how the function behaves for both positive and negative values of \(x\). You see significant growth for positive \(x\) and a flattening curve as \(x\) becomes more negative. This method helps in visualizing the exponential behavior of the function smoothly.

Asymptotes

An asymptote of a function is a line that the graph of the function approaches but never touches.
For an exponential function like \(f(x) = e^{2x}\), the horizontal line \(y = 0\) serves as the asymptote.
This signifies that as \(x\) reduces and approaches negative infinity, \(f(x)\) gets infinitely closer to \(y=0\), but it never actually reaches zero.
Asymptotes are essential in understanding the boundary behavior of graphs:

• They guide the sketching of the graph, ensuring it never crosses certain regions.
• They help visualize limits. In this case, \( e^{2x} \) approaches zero but never becomes negative.

Understanding asymptotes ensures accurate representation of the graph's behavior near the axes. It encapsulates the exponential function's behavior at the extremities.