Problem 30 - Intermediate Algebra

Question

Graph each function. $$ y=e^{x-5} $$

Step-by-Step Solution

Answer

The graph of \(y = e^{x-5}\) is an exponential curve shifted 5 units right, passing through points like (5, 1).

Step 1: Understand the Function

The given function is an exponential function of the form \(y = e^{x-5}\). The base \(e\) is the Euler's number, approximately equal to 2.718. The expression \(x - 5\) in the exponent implies a horizontal shift to the right by 5 units.

Step 2: Identify the Transformation

The function \(y = e^x\) is shifted to the right by 5 units. The point \((x, y) = (5, 1)\) is on the graph because substituting \(x = 5\) we get \(y = e^{5-5} = e^0 = 1\). Thus, the basic shape remains the same, but every point on \(y = e^x\) is moved 5 units to the right.

Step 3: Plot Key Points

Calculate a few key points based on the transformation: - When \(x = 4\), \(y = e^{4-5} = e^{-1} \approx 0.368\)- When \(x = 5\), \(y = e^{5-5} = 1\)- When \(x = 6\), \(y = e^{6-5} = e = 2.718\)These key points help in sketching the graph accurately.

Step 4: Sketch the Graph

Plot the points \((4, 0.368)\), \((5, 1)\), and \((6, 2.718)\) on a coordinate plane. Draw a smooth curve through these points, extending the curve for other points to show the typical exponential growth as \(x\) increases and approaching zero as \(x\) decreases.

Key Concepts

Understanding Horizontal Shifts in Exponential Functions

In an exponential function like \(y = e^{x-5}\), the exponent includes \(x - 5\). The subtraction inside the exponent indicates a horizontal shift. Specifically, since it is \(-5\), the graph of the base exponential function \(y = e^x\) is shifted to the right by 5 units.

• Think of horizontal shift as moving the entire graph to the right or left along the x-axis.
• In this case, every point on the graph of \(y = e^x\) moves 5 units to the right.
• The point \((0, 1)\) on the graph of \(y = e^x\) becomes \((5, 1)\) for \(y = e^{x-5}\).

Visualizing this shift helps you understand the effect of changes within the exponent on the graph's position. Remember, since the shift is horizontal, the overall shape of the function remains an exponential curve; it merely relocates.

Euler's Number and Its Role in Exponential Functions

Euler's number, denoted as \(e\), is a fundamental constant approximately equal to 2.718. It serves as the base of the natural logarithm and is integral to the study of exponential growth and decay.

• Euler's number \(e\) is irrational, meaning it cannot be expressed as a simple fraction.
• The exponential function \(y = e^x\) is used widely in mathematics, particularly in modeling real-world phenomena like population growth and radioactive decay.
• In functions like \(y = e^{x-5}\), \(e\) contributes to the natural growth rate of the function.

By understanding the properties of \(e\), you gain deeper insight into why exponential functions have their unique curves, characterized by rapid increases or decreases.

Determining and Using Key Points for Graphing

In graphing, identifying key points is crucial for drawing an accurate exponential function. For the function \(y = e^{x-5}\), we derive these points by choosing various \(x\)-values and calculating corresponding \(y\)-values.

• A key point, \((5, 1)\), confirms the right shift since plugging \(x = 5\) gives \(y = 1\).
• Choosing \(x = 4\) and \(x = 6\) gives points \((4, 0.368)\) and \((6, 2.718)\) respectively.
• These calculated points help sketch the graph, showing how it approaches zero for \(x < 5\) and grows rapidly for \(x > 5\).

Using key points like these allows for a more comprehensive understanding and visualization of the shifted exponential curve, as they offer exact locations through which the curve passes.