Complete The Table To Investigate Dilations Of Exponential Functions

Investigating Dilations of Exponential Functions: A Comprehensive Guide

Exponential functions are fundamental in mathematics and numerous applications, from modeling population growth to understanding radioactive decay. Understanding how dilations affect these functions is crucial for accurate modeling and interpretation. This article provides a comprehensive exploration of dilations applied to exponential functions, complete with illustrative examples and practical applications.

Understanding Exponential Functions and Dilations

Before diving into dilations, let's refresh our understanding of exponential functions and the concept of dilation itself.

The Basic Exponential Function

The general form of an exponential function is:

f(x) = ab^x

where:

• 'a' represents the initial value or vertical scaling factor. It determines the y-intercept (the value of the function when x=0).
• 'b' is the base, a positive constant greater than 0 and not equal to 1. It determines the rate of growth or decay. If b > 1, the function represents exponential growth; if 0 < b < 1, it represents exponential decay.
• 'x' is the independent variable, often representing time or some other measure.

Dilations: Stretching and Compressing

A dilation is a transformation that stretches or compresses a function. In the context of exponential functions, we can apply dilations vertically or horizontally.

• Vertical Dilation: This stretches or compresses the function along the y-axis. It's achieved by multiplying the entire function by a constant (let's call it 'k'). A vertical dilation of f(x) by a factor of 'k' results in the function kf(x). If |k| > 1, the function stretches vertically; if 0 < |k| < 1, it compresses vertically.

• Horizontal Dilation: This stretches or compresses the function along the x-axis. It's achieved by multiplying the exponent 'x' by a constant (let's call it 'c'). A horizontal dilation of f(x) by a factor of 'c' results in the function f(cx). If |c| > 1, the function compresses horizontally; if 0 < |c| < 1, it stretches horizontally. Note that the effects are opposite to what one might intuitively expect.

Investigating Dilations Through Tables and Graphs

Let's explore the effects of dilations using tables and graphs. We'll use the base exponential function f(x) = 2^x as our starting point.

Example 1: Vertical Dilation

Consider the function f(x) = 2^x. Let's investigate the vertical dilations by factors of 2 and 1/2.

x	f(x) = 2<sup>x</sup>	2f(x)	(1/2)f(x)
-2	0.25	0.5	0.125
-1	0.5	1	0.25
0	1	2	0.5
1	2	4	1
2	4	8	2
3	8	16	4

Observation: Multiplying f(x) by 2 stretches the graph vertically, while multiplying by 1/2 compresses it vertically. The y-values are scaled by the dilation factor.

Example 2: Horizontal Dilation

Now let's consider horizontal dilations of f(x) = 2^x by factors of 2 and 1/2. Remember that the effect is opposite to the intuitive expectation.

x	f(x) = 2<sup>x</sup>	f(2x)	f(x/2)
-2	0.25	0.5	0.353
-1	0.5	1	0.707
0	1	1	1
1	2	4	1.414
2	4	16	2
3	8	64	2.828

Observation: f(2x) compresses the graph horizontally, while f(x/2) stretches it horizontally. The x-values are scaled inversely by the dilation factor.

Combining Vertical and Horizontal Dilations

We can combine vertical and horizontal dilations. Consider the general form:

g(x) = k * a * b^cx

where:

• 'k' is the vertical dilation factor.
• 'a' is the initial value.
• 'b' is the base.
• 'c' is the horizontal dilation factor.

This function represents a vertically dilated and horizontally dilated version of the basic exponential function f(x) = ab^x.

Real-World Applications of Dilated Exponential Functions

Dilated exponential functions are used extensively in various fields:

Population Growth and Decay

Consider a population initially at 1000 individuals with a growth rate of 2% per year. The basic model is:

P(t) = 1000 * (1.02)^t

If the growth rate changes to 4% per year, the model becomes:

P(t) = 1000 * (1.04)^t (Horizontal dilation – it grows faster, compressing horizontally)

If the initial population doubles, we get:

P(t) = 2000 * (1.02)^t (Vertical dilation – the whole function scales vertically)

Radioactive Decay

Radioactive decay follows an exponential decay model. Let's say a substance decays with a half-life of 10 years, starting with 100 grams.

A(t) = 100 * (1/2)^t/10

If we change the half-life to 5 years:

A(t) = 100 * (1/2)^t/5 (Horizontal dilation; decays faster)

If the initial amount is 200 grams:

A(t) = 200 * (1/2)^t/10 (Vertical dilation)

Compound Interest

Compound interest is another classic example. The formula for compound interest is:

A = P (1 + r/n)^nt

where:

• A = the future value of the investment/loan, including interest.
• P = the principal investment amount (the initial deposit or loan amount).
• r = the annual interest rate (decimal).
• n = the number of times that interest is compounded per year.
• t = the number of years the money is invested or borrowed for.

Changes in 'P' (principal) represent vertical dilation, while changes in 'r' (interest rate) or 'n' (compounding frequency) affect the exponential growth itself leading to effective horizontal dilation. A higher interest rate means faster growth and horizontal compression. More frequent compounding also leads to a slightly faster growth (though the effect is less pronounced than changing the interest rate directly).

Conclusion

Understanding the effects of dilations on exponential functions is essential for accurate modeling and interpretation in numerous fields. By manipulating the vertical and horizontal dilation factors, we can adapt the basic exponential function to fit various real-world scenarios involving growth, decay, and other phenomena that follow an exponential pattern. The examples provided above serve as a foundation for further exploration and application in your studies and professional endeavors. Remember to visualize these effects through both tables and graphs for a complete understanding. The interplay between vertical and horizontal dilations opens the door to a more nuanced and accurate representation of complex exponential relationships.