Problem 17 - Introductory and Intermediate Algebra for College Students 4th

Question

Graph each function by making a table of coordinates. If applicable, use a graphing unility to confirm your hand-drawn graph. $$f(x)=4^{x}$$

Step-by-Step Solution

Answer

The plotted points for the function \(f(x) = 4^{x}\) are (-2, 0.0625), (-1, 0.25), (0, 1), (1, 4), and (2, 16). It forms an increasing exponential graph.

Step 1: Constructing a Table of Values

First step is to choose a range of x-values. A recommended option would be between -2 and 2 including the zero. This gives a total of five x-values: -2, -1, 0, 1, 2. Now, compute the corresponding y-values using the function \(f(x) = 4^{x}\).

Step 2: Calculation of Y-Values

Compute the y-values as follows:\n\n For x = -2, \(f(x) = 4^{-2} = 0.0625\)\n\n For x = -1, \(f(x) = 4^{-1} = 0.25\)\n\n For x = 0, \(f(x) = 4^{0} = 1\)\n\n For x = 1, \(f(x) = 4^{1} = 4\)\n\n For x = 2, \(f(x) = 4^{2} = 16\)

Step 3: Plotting the Points

Now plot these points on the graph. The points to be plotted are (-2, 0.0625), (-1, 0.25), (0, 1), (1, 4), and (2, 16).

Step 4: Drawing the graph

Upon plotting, connect these points smoothly to form a curve of the function \(f(x) = 4^{x}\). Notice that this is an increasing exponential graph.

Step 5: Confirm with Graphing Utility

If applicable, input the function \(f(x) = 4^{x}\) in a graphing utility, to confirm the accuracy of the hand-drawn graph.

Key Concepts

Exponential Growth

Exponential growth occurs when a quantity increases at a consistent rate over equal intervals of time. In the context of the function \(f(x) = 4^x\), this means that as \(x\) increases by 1, the value of \(f(x)\) is multiplied by 4. This results in a rapid increase, clearly seen in the sequence of points: from \(0.0625\) when \(x = -2\) to \(16\) when \(x = 2\).
Exponential growth is particularly noteworthy in natural phenomena and finance, such as population growth and compound interest. Understanding how exponential functions grow can help manage expectations and predictions regarding real-world scenarios. Take note that as \(x\) decreases, the values get infinitesimally smaller but never reach zero.

Coordinate Plane

The coordinate plane is a two-dimensional plane where each point is determined by a pair of numerical coordinates. These coordinates are written in the form \((x, y)\), where \(x\) represents the horizontal axis and \(y\) represents the vertical axis.
In graphing the function \(f(x) = 4^x\), we must plot points such as \((-2, 0.0625)\) and \((2, 16)\) on the coordinate plane. The x-values range from negative to positive, illustrating the function behavior over both axes.

• X-axis: The horizontal line where \(y = 0\).
• Y-axis: The vertical line where \(x = 0\).

Crossing the axes at the origin \((0,0)\), the coordinate plane allows us to visually interpret and connect these points to form curves and lines representing functions.

Graphing Utilities

Graphing utilities are tools used to visualize mathematical functions on a coordinate plane. They can be physical devices like graphing calculators or software applications on computers and mobile devices.
When graphing the function \(f(x) = 4^x\), a graphing utility provides an accurate depiction of the exponential curve, confirming what was manually plotted. By entering the function into the utility, a smooth curve that connects the plotted points will appear, illustrating how the curve continues beyond the chosen points.
These utilities are essential in mathematics and education for checking work and understanding the visual representation of complex functions. They help students and professionals save time while reducing errors in graph plotting.

Function Tables

Function tables are useful tools for evaluating and organizing the output of a function for various inputs. When creating a function table for \(f(x) = 4^x\), you choose several \(x\)-values and calculate the respective \(f(x)\) values, resulting in ordered pairs like \((-2, 0.0625)\) and \((2, 16)\).
Using a function table facilitates predictability and structure, showing how the value of \(f(x)\) shifts as \(x\) changes. This organization is key for plotting graphs or solving larger algebraic problems. Such tables allow for easy cross-referencing, ensuring that the corresponding points are correctly plotted on a graph. They also aid in understanding the function's pattern and growth behavior.