Problem 47 - Intermediate Algebra

Question

For Problems \(35-52\), graph each exponential function. $$ f(x)=3^{2 x} $$

Step-by-Step Solution

Answer

Plot points using f(x) = 3^{2x}, draw the curve, noting exponential growth.

Step 1: Understand the Exponential Function

The function given is an exponential function in the form of \( f(x) = a^{bx} \), where \( a = 3 \) and \( b = 2 \). This means our function is \( f(x) = 3^{2x} \).

Step 2: Create a Table of Values

Choose several values for \( x \), substitute them into \( f(x) = 3^{2x} \), and calculate the corresponding \( f(x) \) values. This will help us plot the graph:- If \( x = -1 \), \( f(x) = 3^{2(-1)} = 3^{-2} = \frac{1}{9} \)- If \( x = 0 \), \( f(x) = 3^{2(0)} = 3^0 = 1 \)- If \( x = 1 \), \( f(x) = 3^{2(1)} = 3^2 = 9 \)- If \( x = 2 \), \( f(x) = 3^{2(2)} = 3^4 = 81 \)

Step 3: Plot the Points on the Graph

Using the points calculated in the table, plot them on a coordinate plane: - \( (-1, \frac{1}{9}) \) - \( (0, 1) \)- \( (1, 9) \)- \( (2, 81) \)These points represent the curve of the function as it progresses.

Step 4: Draw the Exponential Curve

After plotting the points from Step 3, draw a smooth curve starting from left, which approaches the x-axis (but never touches it) and rises steeply through the plotted points. This curve illustrates the exponential growth of the function.

Step 5: Analyze the Graph

Observe that this exponential function graph has a horizontal asymptote at \( y = 0 \), indicating the graph never touches the x-axis. As \( x \) increases, \( f(x) \) rises sharply, reflecting exponential growth.

Key Concepts

Exponential Growth

Exponential growth occurs when a quantity increases by a constant factor over equal increments of time or space. In the context of graphing exponential functions, it means that as the variable increases, the value of the function rises at a faster and faster rate. This rapid increase is characteristic of functions with the general form \( f(x) = a^{bx} \).

• In our specific example, \( f(x) = 3^{2x} \), the base \( a = 3 \) determines how quickly the function grows. As \( x \) increases by 1, \( f(x) \) is multiplied by \( 3^2 = 9 \).
• The substance of exponential growth can be visualized on a graph: starting relatively flat and shifting to a steep, upward curve.

Understanding exponential growth is essential for recognizing how small changes in \( x \) result in large changes in \( f(x) \) as the graph continues. This is why it appears to suddenly "shoot up" after a certain point. It’s an incredibly powerful concept frequently encountered in real-world applications, such as population growth or compound interest.

Asymptote

An asymptote refers to a line that a curve approaches infinitely closely but never actually intersects. In the realm of exponential functions, identifying asymptotes helps understand the behavior of the graph as the independent variable \( x \) moves toward positive or negative infinity.

• For the function \( f(x) = 3^{2x} \), the horizontal asymptote is \( y = 0 \). This reflects the fact that as \( x \) moves towards negative infinity, \( f(x) \) approaches zero but never actually reaches it.
• It signifies that the values of \( f(x) \) become very small but remain positive, never true zero.

Future behavior prediction of the graph often involves asymptotes, as they set boundaries the graph will move under or above. For students learning to graph exponential functions, identifying an asymptote provides insight into the long-term trend of the graph.

Coordinate Plane

The coordinate plane is the framework where functions are visually represented. It consists of two perpendicular axes – the x-axis (horizontal) and the y-axis (vertical) – that allow for the graphical understanding of mathematical functions. By plotting points, we can visually interpret how functions behave.

• For our function \( f(x) = 3^{2x} \), crucial points were plotted on this plane: \((-1, \frac{1}{9})\), \((0, 1)\), \((1, 9)\), and \((2, 81)\).
• Each point corresponds to a pair \( (x, f(x)) \), showing how changing \( x \) results in changes to \( f(x) \).

This visual representation is crucial when dealing with exponential growth, as it offers an intuitive understanding of the effect of exponential change. The coordinate plane helps students see these effects clearly, making the abstract concept of exponential functions tangible.

Table of Values

Creating a table of values is a methodical way to see how the output of a function changes as its input values change. It simplifies the process of plotting a graph for complex functions like exponential ones.

• For \( f(x) = 3^{2x} \), we chose specific values of \( x \) – such as -1, 0, 1, and 2 – to calculate corresponding \( f(x) \) values.
• The evaluated points were: \((-1, \frac{1}{9})\), \((0, 1)\), \((1, 9)\), \((2, 81)\).
• This approach can also predict behavior between these points by visualizing their connections as segments of the curve.

Using a table of values provides clarity and structure to the process of graphing, ensuring each point is accurately calculated before plotting. For educators, it’s a crucial step in demonstrating the tangible impact \( x \) values have on \( f(x) \), deepening student understanding of exponential functions.