Problem 58 - College Algebra

Question

Use point plotting to graph \(f(x)=2^{x}\). Begin by setting up a partial table of coordinates, selecting integers from \(-3\) to 3 , inclusive, for \(x\). Because \(y=0\) is a horizontal asymptote, your graph should approach, but never touch, the negative portion of the \(x\) -axis.

Step-by-Step Solution

Answer

Plot the points \((-3, 0.125), (-2, 0.25), (-1, 0.5), (0, 1), (1, 2), (2, 4), (3, 8)\) on a graph, then draw a smooth curve through these points showing that as \(x\) becomes more negative, the function \(f(x)\) approaches 0 but never becomes 0.

Step 1: Understand the function

The given function is \(f(x)=2^x\). It's an exponential function which increases as x grows. When \(x=0, f(x)=1\); as \(x\) moves towards positive infinity, \(f(x)\) increases, and as \(x\) decreases or moves towards negative infinity, \(f(x)\) approaches 0 but never becomes 0.

Step 2: Set up a table of values

Set up a table with two columns, \(x\) and \(f(x)\). Fill in the \(x\) column with values from -3 to 3. These values should include -3, -2, -1, 0, 1, 2, and 3.

Step 3: Calculate corresponding y-values

Use the function \(f(x)=2^x\) to calculate the values of \(f(x)\) for each selected \(x\). So when \(x = -3, f(x) = 2^{-3} = 0.125\); when \(x = -2, f(x) = 2^{-2} = 0.25\); when \(x = -1, f(x) = 2^{-1} = 0.5\); when \(x = 0, f(x) = 2^{0} = 1\); and so on.

Step 4: Plot the coordinates

Plot the coordinates \(x, f(x)\) on a graph. This can be done by drawing an x-axis and y-axis on a graph, marking off equal intervals, and then placing a point for each \(x, f(x)\) pair from the table.

Step 5: Draw the curve

Draw a smooth curve through all the plotted points.The graph should clearly show that as \(x\) tends towards negative infinity, the graph of \(f(x)\) approaches but never touches the negative x-axis, which is the horizontal asymptote.

Key Concepts

Point Plotting

Point plotting is a fundamental technique used to visually represent a function on a graph. This technique involves identifying key coordinates derived from the function, and accurately placing them on a graph.
For the function \(f(x) = 2^x\), you'll begin by selecting a range of \(x\)-values, such as integers from -3 to 3. For each \(x\) value, calculate the corresponding \(f(x)\) value. These pairs create a series of specific points.
Next, position these points on your graph, aligning with the \(x\) and \(y\) axes. By doing so, you create a visual map of where the function lies. It's essential to ensure precision when placing points, as these guide the smooth curve you'll eventually draw.
Point plotting not only helps in creating graphs but also deepens your understanding of how functions behave. It offers insights into the rate of change within the function and how it responds to different inputs.

Coordinates Table

Utilizing a coordinates table is a streamlined approach to organize points before plotting them. Each row in the table represents a specific \(x, f(x)\) pair, making it easy to systematically transition from calculation to graphing.
In our example with \(f(x) = 2^x\), you would start by listing \(x\) values ranging from -3 to 3 in one column. In the adjacent column, calculate and record the corresponding \(f(x)\) values using the function equation.

• For \(x = -3\), \(f(x) = 2^{-3} = 0.125\)
• For \(x = -2\), \(f(x) = 2^{-2} = 0.25\)
• For \(x = -1\), \(f(x) = 2^{-1} = 0.5\)
• And so on, until \(x = 3\)

This tabular form not only aids in preventing calculation errors but also offers a quick reference when plotting the graph, helping you visualize the growth or decay of the function effectively.

Horizontal Asymptote

A horizontal asymptote is a horizontal line that the graph of the function approaches as \(x\) approaches positive or negative infinity. In the context of the function \(f(x) = 2^x\), the horizontal asymptote is significant because it visually represents the boundary that the exponential graph will approach but never touch.
For the function \(f(x) = 2^x\), the horizontal asymptote is the line \(y = 0\). As \(x\) decreases and moves towards negative infinity, \(f(x)\) gets closer and closer to zero, but it never actually becomes zero. This is due to the nature of exponential decay; the values get infinitesimally small but remain positive.
Recognizing the presence of a horizontal asymptote helps to maintain an accurate sketch of the graph. As you draw the curve, ensure to let it swoop closer to, but not intersect, the horizontal line of the asymptote.

Graphing Functions

Graphing an exponential function like \(f(x) = 2^x\) involves multiple steps, all of which contribute to a complete and accurate representation of the function's behavior. Once you've plotted the points using a coordinates table, the next step is to sketch a smooth curve connecting them.
Begin by marking out the x-axis and y-axis on graph paper or a digital tool, ensuring equal spacing for optimum precision. After plotting all coordinates, carefully draw a curve that smoothly interconnects these points. Pay attention to how the curve behaves:

• As \(x\) heads towards positive infinity, the function value increases sharply, exemplifying the exponential growth.
• Conversely, as \(x\) slides towards negative infinity, verify that the curve flattens out near the horizontal asymptote, \(y=0\).

Graphing serves a dual purpose. It not only offers a visual representation of abstract mathematical functions but also facilitates deeper comprehension of the function’s intrinsic characteristics and potential boundaries.