Problem 25 - Brief Applied Calculus

Question

$$ \text { Graph each function } $$ $$ f(x)=3^{x} $$

Step-by-Step Solution

Answer

Graph \( f(x) = 3^x \) as a smooth curve passing through key points; it has a horizontal asymptote at \( y = 0 \).

Step 1: Understand the Function

The function given is \( f(x) = 3^x \). This is an exponential function with a base of 3. In exponential functions of the form \( a^x \), the graph rises from left to right because the base is greater than 1.

Step 2: Plot Key Points

Choose several values for \( x \) to plot key points that will help sketch the graph accurately. Calculate \( f(x) \) for \( x = -2, -1, 0, 1, ext{and } 2 \):- \( f(-2) = 3^{-2} = \frac{1}{9} \)- \( f(-1) = 3^{-1} = \frac{1}{3} \)- \( f(0) = 3^0 = 1 \)- \( f(1) = 3^1 = 3 \)- \( f(2) = 3^2 = 9 \)

Step 3: Graph the Points

On a coordinate plane, plot the points calculated from Step 2: - (-2, \( \frac{1}{9} \))- (-1, \( \frac{1}{3} \))- (0,1)- (1,3)- (2,9).These points help form the general shape of the exponential graph.

Step 4: Draw the Curve

Draw a smooth curve through the plotted points that extends infinitely in the positive \( y \) direction as \( x \) increases and approaches the x-axis (but never touches it) as \( x \) decreases. This curve will represent the graph of \( f(x) = 3^x \).

Step 5: Identify Asymptote and Behavior

Identify that the x-axis (\( y = 0 \)) is a horizontal asymptote; the graph approaches but never touches or crosses the x-axis. As \( x \) approaches positive infinity, \( f(x) \) increases rapidly. As \( x \) approaches negative infinity, \( f(x) \) approaches zero.

Key Concepts

Graphing Exponential Functions

Graphing exponential functions like \( f(x) = 3^x \) involves plotting key points and understanding the overall shape of the graph. An exponential function generally takes the form \( a^x \), where \( a \) is a constant, and \( x \) is the exponent. If \( a > 1 \), the graph of \( a^x \) will rise from left to right, creating a curve that starts close to the x-axis on the left and rises rapidly on the right.
To begin graphing \( f(x) = 3^x \), you choose a few critical values of \( x \)—for instance, \(-2, -1, 0, 1,\) and \(2\). For each of these \( x \) values, calculate corresponding \( f(x) \):

• \( f(-2) = 3^{-2} = \frac{1}{9} \)
• \( f(-1) = 3^{-1} = \frac{1}{3} \)
• \( f(0) = 3^0 = 1 \)
• \( f(1) = 3^1 = 3 \)
• \( f(2) = 3^2 = 9 \)

Once these points are determined, plot them on a coordinate plane. You will notice that the points will start close to the x-axis on the negative side of the \( x \)-axis and climb steeply upward as \( x \) becomes positive.

Properties of Exponential Graphs

Exponential graphs, such as \( f(x) = 3^x \), possess unique properties that are crucial to understand. One of the most notable features of these graphs is their rapid change rate, which increases for larger \( x \) values. This results in a sharp curve that becomes steeper as \( x \) grows. These graphs are neither linear nor symmetrical.
Key characteristics of exponential graphs include:

• Exponential Growth: When the base \( a \) is greater than 1, the function exhibits exponential growth, meaning the function’s value rapidly increases as \( x \) increases.
• Y-intercept: The graph of an exponential function \( a^x \) with any base will intersect the y-axis at 1, because \( a^0 = 1 \).
• Unbounded: These graphs extend infinitely upwards, without any horizontal bounds as \( x \rightarrow \)

Understanding these features allows you to predict the general behavior of exponential functions and tailor their representation accurately on a graph.

Asymptotes in Functions

An essential aspect of graphing exponential functions is recognizing asymptotes, which are lines that a graph approaches but never touches. In the case of \( f(x) = 3^x \), the x-axis, or \( y = 0 \), is a horizontal asymptote. As \( x \) becomes more negative (\( x \to -\infty \)), the values of \( 3^x \) become very small, approaching zero, but never actually reaching it.
This horizontal asymptote indicates that, although the curve comes close, it shall never intersect or cross the x-axis. As \( x \) increases positively, \( 3^x \) grows without any upper bound, distancing itself further from the asymptote. Understanding the behavior of exponential graphs near their asymptotes helps in sketching these curves accurately. It ensures that you know where to draw the graph without misrepresentation of its behavior near these critical lines.