Problem 2 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch a graph of the given exponential function. $$ f(x)=\frac{1}{3} 5^{x} $$

Step-by-Step Solution

Answer

The graph of \( f(x) = \frac{1}{3} \cdot 5^x \) is an exponential growth curve with a y-intercept at \( y = \frac{1}{3} \), rising steeply as \( x \) increases.

Step 1: Identify the Function Type

The function given is \( f(x) = \frac{1}{3} \cdot 5^x \). This is an exponential function, where the base is 5 and the coefficient in front is \( \frac{1}{3} \). Exponential functions have the general form \( f(x) = a \cdot b^x \), where \( a eq 0 \) and \( b > 0 \).

Step 2: Determine Key Characteristics

The exponential function \( f(x) = \frac{1}{3} \cdot 5^x \) has several key characteristics: 1. The base (5) is greater than 1, indicating growth.2. The coefficient (\( \frac{1}{3} \)) is positive, affecting the vertical stretch and the initial value.3. Its horizontal asymptote is at \( y = 0 \) because the function approaches zero as \( x \to -\infty \).

Step 3: Calculate Intercepts and Select Points

To sketch the graph, calculate intercepts and additional points:- Y-intercept: When \( x = 0 \), \( f(0) = \frac{1}{3} \cdot 5^0 = \frac{1}{3} \).- Select a few more values for \( x \) to get points on the graph: - \( x = 1 \): \( f(1) = \frac{1}{3} \cdot 5 = \frac{5}{3} \). - \( x = -1 \): \( f(-1) = \frac{1}{3} \cdot 5^{-1} = \frac{1}{3 \times 5} = \frac{1}{15} \).

Step 4: Sketch the Graph

Using the calculated points, sketch the graph:1. Plot the y-intercept at \( (0, \frac{1}{3}) \).2. Plot \( (1, \frac{5}{3}) \) showing the function is growing to the right.3. Plot \( (-1, \frac{1}{15}) \) showing the function is decreasing to zero to the left.4. Draw the curve smoothly passing through these points, showing an increase with steepness to the right.

Step 5: Note Graph Features

The graph of \( f(x) = \frac{1}{3} \cdot 5^x \) is an increasing curve that passes through \( (0, \frac{1}{3}) \) and rises steeply as \( x \) increases. It approaches the horizontal asymptote \( y = 0 \) as \( x \) decreases. There are no x-intercepts as the function never touches the x-axis.

Key Concepts

Graph Sketching

When sketching the graph of an exponential function, it helps to first understand the basic form and characteristics of the function. Exponential functions are typically expressed as \( f(x) = a \cdot b^x \), where \( a \) determines the vertical stretch or compression, and \( b \), the base, dictates the growth or decay rate. For the function \( f(x) = \frac{1}{3} \cdot 5^x \), these elements reveal that the graph will

• Start at the point determined by \( a \), which is \( \frac{1}{3} \) here.
• Exhibit exponential growth since \( b=5 \) is greater than 1.

To visualize this function, plot the y-intercept and other key points calculated using several \( x \) values. Then, draw a smooth curve through these points, ensuring the curve gets steeper as \( x \) increases. This methodical approach ensures a clear and accurate sketch of the function.

Asymptotes

Asymptotes are lines that the graph of the function approaches but never touches. They are significant in understanding the end behavior of a function. For \( f(x) = \frac{1}{3} \cdot 5^x \), the horizontal asymptote is at \( y = 0 \). This tells us that as \( x \) moves towards negative infinity, the value of \( f(x) \) will get closer and closer to zero, but it will never actually reach zero.
Exponential functions typically have only a horizontal asymptote (unless transformed otherwise), providing insight into how the function behaves over the long term. In graph sketching, this asymptote guides the portion of the curve that decreases towards zero, ensuring it stays above the x-axis, hovering close to it as \( x \) heads towards negative infinity.

Intercepts

Intercepts are pivotal points where the graph crosses the axes. For the given function, we determine intercepts by setting \( x \) or \( f(x) \) to zero and solving for the remaining variable.

• Y-intercept: This occurs where the graph crosses the y-axis, at \( x = 0 \). For \( f(x) = \frac{1}{3} \cdot 5^x \), when \( x = 0 \), \( f(0) = \frac{1}{3} \), making the y-intercept at the point \( (0, \frac{1}{3}) \).
• X-intercept: This function does not have an x-intercept. Since exponential functions of this form (with \( a > 0 \)) never become zero, the graph will not intersect the x-axis.

Plotting these intercepts gives us foundational points for sketching the function's graph.

Function Characteristics

Understanding the unique traits of exponential functions helps in predicting and contextualizing their graphs. The function \( f(x) = \frac{1}{3} \cdot 5^x \) showcases the following important characteristics:

• Growth Rate: The base, 5, indicates that this is a growth function. Each increment in \( x \) multiplies the output by 5, creating a steep ascending curve.
• Initial Value Influence: The coefficient \( \frac{1}{3} \) impacts the rate at which the function starts growing, compressing the curve vertically compared to higher coefficients.
• Continuity and Smoothness: Exponential functions are continuous and smooth, lacking abrupt breaks or corners. This quality simplifies their graphing as smooth, continuous curves.

These characteristics ensure that the function starts above zero, grows rapidly, and smooths into an elegant curve. Understanding these gives students a clearer picture of how exponential graphs behave.