Problem 28 - Algebra for College Students

Question

Graph each of the exponential functions. $$ f(x)=4^{x} $$

Step-by-Step Solution

Answer

Graph the function by plotting points and sketching a curve for \( f(x) = 4^x \).

Step 1: Identify the base and the function

The given function is \( f(x) = 4^x \). The base of the exponential function is 4, which means the function will grow rapidly as x increases. The general form of an exponential function is \( f(x) = a^x \), where "a" is the base.

Step 2: Create a table of values

To graph the function, we first create a table of values by selecting some x-values and calculating corresponding y-values using \( f(x) = 4^x \).\[\begin{array}{|c|c|}\hlinex & f(x) \\hline-2 & \frac{1}{16} \-1 & \frac{1}{4} \0 & 1 \1 & 4 \2 & 16 \\hline\end{array}\]

Step 3: Plot the points on the graph

Using the table of values, plot the points \((-2, \frac{1}{16})\), \((-1, \frac{1}{4})\), \((0, 1)\), \((1, 4)\), and \((2, 16)\) on a coordinate plane. These points represent the function's possible values.

Step 4: Sketch the graph

Connect the plotted points with a smooth curve to represent the function \( f(x) = 4^x \). The graph will be an exponentially increasing curve, starting near the x-axis (as \( x \) becomes very negative) and rising steeply as \( x \) becomes positive.

Key Concepts

Graphing Functions

Graphing exponential functions, such as \( f(x) = 4^x \), involves understanding how the function behaves as the value of \( x \) changes. Exponential functions have a constant base, in this case 4, raised to the power of \( x \). This means that as \( x \) increases, the value of \( f(x) \) grows rapidly. For exponential functions, the pattern of growth is multiplicative, which results in a characteristic curve on the graph.
To graph these functions, it's important to first determine key points where the function's behavior is most evident. Generally, starting with a table of values can provide a visual blueprint for plotting these points onto a graph.

Once plotted on a coordinate plane, these points can be connected to form a smooth, continuous curve known as the exponential curve. This curve, when moving left to right, will always start closer to, but never touching, the x-axis for negative \( x \)-values, then rise steeper with positive \( x \)-values, indicating exponential growth.

• The graph will be increasingly steep as \( x \) becomes more positive.
• The function will never touch the x-axis, illustrating that exponential functions never actually reach zero.

Table of Values

Creating a table of values is crucial for graphing exponential functions accurately. This step involves selecting a set of \( x \)-values, often ranging from negative to positive integers, and calculating the corresponding \( f(x) \) values using the exponential equation. In the function \( f(x) = 4^x \), each \( x \) generates a different value of \( f(x) \) based on the power raised.

For example:

• When \( x = -2 \), \( f(x) = 4^{-2} = \frac{1}{16} \).
• When \( x = -1 \), \( f(x) = 4^{-1} = \frac{1}{4} \).
• When \( x = 0 \), \( f(x) = 4^0 = 1 \).
• When \( x = 1 \), \( f(x) = 4^1 = 4 \).
• When \( x = 2 \), \( f(x) = 4^2 = 16 \).

These points help establish a clear representation of how the function behaves across different sections of the graph. The table essentially serves as a guide to plot these points on the coordinate plane, ensuring an accurate graphical representation of the function. By examining the table, you can observe how the \( y \)-values increase exponentially as \( x \) increases from negative to positive.

Coordinate Plane

The coordinate plane is a two-dimensional space for plotting mathematical functions. It consists of two perpendicular axes: the horizontal x-axis and the vertical y-axis. When graphing the function \( f(x) = 4^x \), the coordinate plane becomes the area where you place your calculated points from the table of values.

To properly plot the points on a coordinate plane:

• Start by identifying the scale on each axis. Make sure the x-axis and y-axis can accommodate both negative and positive values, given the exponential nature of the function.
• Place each point accordingly. For example, the point \((0, 1)\) lies directly on the y-axis, representing the y-intercept of the function.
• Ensure equal spacing between incremental units for accuracy.

The plotting might look something like this: starting with the point \((-2, \frac{1}{16})\) very close to the x-axis, you progress to \((2, 16)\) higher up on the y-axis, illustrating how sharply exponential functions rise.
By understanding how to use the coordinate plane effectively, graphing functions like those in exponential form become intuitive, providing insight into their behavior and properties. This visual representation helps in grasping how exponential growth functions develop.