Problem 27 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Sketch the graphs of the given functions on the same axes. \(y=4^{0.5 x}, y=4^{x}\), and \(y=4^{2 x}\)

Step-by-Step Solution

Answer

Begin by drawing a set of axes. Plot a few points for each function: \(y = 4^{0.5x}\), \(y = 4^x\), and \(y = 4^{2x}\), then connect them smoothly to sketch the graphs of these functions. Observe that all graphs pass through point (0,1) and as the constant inside the exponent increases, the graph becomes steeper. Thus, \(y = 4^{0.5x}\) grows slower than \(y = 4^x\), which in turn grows slower than \(y = 4^{2x}\).

Step 1: Set up the axes

Begin by drawing a set of axes. Label the horizontal axis as x and the vertical axis as y.

Step 2: Plot the first function, y = \(4^{0.5x}\)

Start by plotting a few points for the first function, y = \(4^{0.5x}\). Some points to consider are (-6, 1/64), (-4, 1/16), (-2, 1/4), (0,1), (2,4), and (4,16). Connect these points smoothly to sketch the graph of this function. Label the curve with the function's equation to help differentiate it from the others.

Step 3: Plot the second function, y = \(4^x\)

Next, plot a few points for the second function: y = \(4^x\). Some points to consider are (-2, 1/16), (-1, 1/4), (0,1), (1,4), and (2,16). Connect these points smoothly to sketch the graph of this function. Label the curve with the function's equation to help differentiate it from the others.

Step 4: Plot the third function, y = \(4^{2x}\)

Lastly, plot a few points for the third function: y = \(4^{2x}\). Some points to consider are: (-1, 1/16), (-0.5, 1/4), (0,1), (0.5,4), and (1,16). Connect these points smoothly to sketch the graph of this function. Label the curve with the function's equation to help differentiate it from the others.

Step 5: Compare the graphs

Observe that all three functions pass through the point (0,1), since any number raised to the power of 0 is 1. As the constant inside the exponent increases, the graph becomes steeper, indicating a faster rate of growth. The graph of y = \(4^{0.5x}\) grows slower than y = \(4^x\), which in turn grows slower than y = \(4^{2x}\).

Key Concepts

Graphing Exponential Functions

In mathematics, graphing exponential functions allows us to visualize how these functions behave under various conditions. When dealing with exponential functions such as \(y=4^{0.5x}\), \(y=4^x\), and \(y=4^{2x}\), there are key patterns to observe in their graphs.

An exponential function is generally of the form \(y = a^{bx}\), where \(a\) is a constant base greater than zero and \(b\) represents a rate determining exponent. Here, the base \(a\) is \(4\), while the exponents \(x\), \(0.5x\), and \(2x\) modify the rate at which the functions grow or decrease.

When we plot these functions:

• They will all pass through the point \((0,1)\) because any number raised to the power of zero is equal to one.
• As we move to the right along the x-axis, you'll notice the functions increasing at different rates, forming curves that move upwards.
• The graph shapes depend on the power: smaller powers like \(0.5x\) result in a slower rise, while larger powers like \(2x\) make the graph increase more steeply.

In essence, graphing these functions provides a visual representation of how rapid or gradual exponential functions expand.

Rate of Growth

When discussing the rate of growth in exponential functions, we are talking about how quickly the value of the function increases as the input \(x\) becomes larger. This aspect is crucial when interpreting changes in exponential expressions.

The function \(y=4^{2x}\) exemplifies a rapid rate of growth. Doubling the exponent makes the function's value increase quickly, because the exponent \(x\) is multiplied by \(2\). On the other hand, for \(y=4^{0.5x}\), the function grows at a slower pace. This happens because we are effectively taking a square root of the base 4 as we increment \(x\).

This varying growth rate among the functions can be attributed to the different powers applied to \(x\):

• Higher exponents lead to steeper curves and faster growth.
• Lower exponents, or fractions of the exponent, slow down the rise, making the graph less steep.

Understanding these differences aids in predicting how exponential functions behave over time, whether concerning populations, finances, or any other growing quantities.

Function Transformation

Function transformation refers to changes made to the basic function \(y = a^x\) that affect its graph in several ways. For exponential functions, transformations often involve altering the coefficient or the exponent.

In the given functions \(y=4^{0.5x}\), \(y=4^x\), and \(y=4^{2x}\), the transformation affects the exponent.

• Scaling the exponent, as seen with the first and third functions, results in vertical compressions or stretches of the graph. \(4^{0.5x}\) represents compression, making the function rise more slowly than \(4^x\).
• On the contrary, \(4^{2x}\) represents an expansion, so the graph rises more steeply—an indicator of rapid growth compared to \(4^x\).

These transformations highlight how exponential functions can adjust to display different growth patterns. Being mindful of exponent changes allows you to foresee and sketch their graphical behavior with greater accuracy.