Problem 47 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Sketch the graphs of the equations on the same coordinate axes. \(y=2^{x}\) and \(y=\log _{2} x\)

Step-by-Step Solution

Answer

To sketch the graphs of \(y = 2^x\) and \(y = \log_2{x}\) on the same coordinate axes: 1. Identify the domain and range of both functions. For \(y = 2^x\), the domain is all real numbers and the range is all positive numbers. For \(y = \log_2{x}\), the domain is all positive numbers and the range is all real numbers. 2. Find key points: \(y = 2^x\) has a \(y\)-intercept at (0,1), and \(y = \log_2{x}\) has an \(x\)-intercept at (1,0). 3. Sketch each graph separately: \(y = 2^x\) rapidly grows in the positive \(x\) direction while \(y = \log_2{x}\) grows unbounded at a decreasing rate. 4. Combine the graphs: Plot both functions on the same axes and notice the intersection at the point (1,1).

Step 1: Identify domain and range

For the exponential function \(y = 2^x\), the domain is all real numbers, and the range is all positive numbers. For the logarithmic function \(y = \log_2{x}\), the domain is all positive numbers, and the range is all real numbers.

Step 2: Find key points

For \(y = 2^x\), we can identify the following key points: - The \(y\)-intercept: when \(x = 0\), \(y = 2^0 = 1\). So the intercept is at the point (0,1). For \(y = \log_2{x}\), we can identify the following key points: - The \(x\)-intercept: when \(y = 0\), \(x = 2^0 = 1\). So the intercept is at the point (1,0).

Step 3: Sketch the individual graphs

For \(y = 2^x\), we have an exponential function with the following characteristics: - Rapid growth in the positive \(x\) direction, - Decreases asymptotically to 0 in the negative \(x\) direction, - Passes through the point (0,1). For \(y = \log_2{x}\), we have a logarithmic function with the following characteristics: - Grows unbounded in the positive \(x\) direction, but at a decreasing rate, - Decreases without bound in the negative \(y\) direction (since the range is all real numbers), - Passes through the point (1,0), - Has an asymptote at \(x = 0\).

Step 4: Combine the graphs on the same coordinate axes

Now we can place both graphs on the same coordinate axes: 1. Draw the exponential function \(y = 2^x\), which rapidly grows in the positive \(x\) direction and asymptotically approaches 0 in the negative \(x\) direction. 2. Draw the logarithmic function \(y = \log_2{x}\), which grows unbounded at a decreasing rate in the positive \(x\) direction, and has an asymptote at \(x = 0\). 3. Notice the intersection of the two graphs at the point (1,1), where \(y = 2^x\) and \(y = \log_2{x}\) are equal. Once you have completed these steps, you will have successfully sketched the graphs of \(y = 2^x\) and \(y = \log_2{x}\) on the same coordinate axes.

Key Concepts

Exponential Functions

Exponential functions are a special class of mathematical functions where a constant base is raised to a variable exponent. In our task, we’re dealing with the function \(y = 2^x\). Here, the base is 2, indicating how the function scales as \(x\) changes.

Key properties of exponential functions include:

• Rapid Growth: As \(x\) increases, \(y = 2^x\) grows very quickly. This is because each increase in \(x\) results in multiplying the previous value of \(y\) by 2.
• Horizontal Asymptote: Like many exponential functions, \(y = 2^x\) has a horizontal asymptote. As \(x\) approaches negative infinity, \(y\) approaches zero but never really reaches it. It gets infinitesimally close to zero.
• Positive Range: Importantly, \(y = 2^x\) only takes positive values, which means the graph will stay above the \(x\)-axis.
• Passes Through Point (0,1): At \(x=0\), the function results in \(y = 2^0 = 1\). This gives us the \(y\)-intercept (0,1) on the graph.

Studying such characteristics helps understand how exponential functions behave and aids in graphing them accurately.

Logarithmic Functions

Logarithms are the inverses of exponential functions. They let us find the exponent we raise a base number to in order to get another number. For example, in the function \(y = \log_2{x}\), we are solving for \(y\) such that \(2^y = x\).

Key properties of logarithmic functions include:

• Inverse Growth: Unlike exponential functions that grow rapidly as \(x\) increases, logarithmic functions grow slowly and sometimes seem to "flatten out." This is why we say they grow unbounded but at a decreasing rate.
• Vertical Asymptote: The function \(y = \log_2{x}\) has a vertical asymptote at \(x = 0\). This is because you cannot take the logarithm of zero or negative numbers.
• All Real Number Range: A major feature is its range, which covers all real numbers. This means \(y = \log_2{x}\) can output any real value if provided with an appropriate positive \(x\) value.
• Passes Through Point (1,0): A special feature of logarithmic functions is the \(x\)-intercept, which is found where the function outputs zero. For \(y = \log_2{x}\), this occurs at the point (1,0), as \(\log_2{1} = 0\).

Understanding these properties is crucial for effectively graphing and analyzing logarithmic functions.

Coordinate Axes

The coordinate axes are fundamental for graphing any function or equation. They consist of two perpendicular lines, the horizontal \(x\)-axis and the vertical \(y\)-axis.

Key points about coordinate axes include:

• Origin: The point where the \(x\)-axis and \(y\)-axis intersect is called the origin, usually denoted as (0, 0). It is the reference point for locating all other points on the graph.
• Positive and Negative Regions: The axes divide the plane into four quadrants. The right of the \(y\)-axis and above the \(x\)-axis are positive, whereas the left and below the axes are negative.
• Units of Measurement: Axes act as scaling references. Points are plotted according to their coordinate values (\(x, y\)). These values determine how far and in which direction a point is from the origin.

Using coordinate axes effectively is key to graphing. In our graphing exercise, plotting functions like \(y = 2^x\) and \(y = \log_2{x}\) involves translating their mathematical properties into visual representations on these axes, allowing us to easily see relationships and interactions.