Problem 2 - Algebra 1: Concepts and Skills

Question

Copy and complete the table of values for the exponential function. $$\begin{array}{|c|c|c|c|c|c|c|}\hline x & {-2} & {-1} & {0} & {1} & {2} & {3} \\\ \hline y=4^{x} & {?} & {?} & {?} & {?} & {?} & {?} \\ \hline\end{array}$$

Step-by-Step Solution

Answer

The completed table is: when x=-2, y=1/16; when x=-1, y=1/4; when x=0, y=1; when x=1, y=4; when x=2, y=16; when x=3, y=64.

Step 1: Identify the Function

In this exercise, we are given a table with a function \(y=4^{x}\). This is an exponential function where the base is 4.

Step 2: Calculate for x=-2

Substitute -2 for x in the function: \(4^{-2} = 1 / 4^{2} = 1/16\). So, the y-value when x=-2 is 1/16.

Step 3: Calculate for x=-1

Substitute -1 for x in the function: \(4^{-1} = 1 / 4^{1} = 1/4\). So, the y-value when x=-1 is 1/4.

Step 4: Calculate for x=0

Substitute 0 for x in the function: \(4^{0}=1\). Any number (except 0) to the power of 0 is 1. So, the y-value when x=0 is 1.

Step 5: Calculate for x=1

Substitute 1 for x in the function: \(4^{1}=4\). So, the y-value when x=1 is 4.

Step 6: Calculate for x=2

Substitute 2 for x in the function: \(4^{2}=16\). So, the y-value when x=2 is 16.

Step 7: Calculate for x=3

Substitute 3 for x in the function: \(4^{3}=64\). So, the y-value when x=3 is 64.

Key Concepts

Exponential Function Table

To grasp the pattern of an exponential function, it's helpful to create a table of values. As in the example of the function y=4^x, filling out a table for various x values provides a clear vision of how the output, often referred to as y, changes exponentially as the input x increases or decreases.

When calculating, you begin by plugging in the simplest value, which is often x=0, because any non-zero base raised to the zero power equals one. Then you proceed by sequentially increasing or decreasing the exponent. This systematic approach allows us to quickly notice the trend: as the exponent grows, the function value increases rapidly if the base is greater than one—this is the essence of exponential growth.

For negative exponents, the function yields a fractional result, which gets smaller as the exponent decreases. For instance, in our exercise, 4^-2 is smaller than 4^-1. Exponential functions demonstrate growth or decay, and making a table serves as a visual tool to understand this concept intuitively.

Calculating Exponential Functions

Calculating values for an exponential function, like y=4^x, involves applying the base of 4 to various power values. This process begins with a clear understanding of what it means to raise a number to a power. When you have a positive exponent, you multiply the base by itself the number of times indicated by the exponent. For example, 4^2 means you multiply 4 by itself once: 4 * 4, which equals 16.

For negative exponents, the calculation involves division. A negative exponent tells us to take the reciprocal of the base raised to the absolute value of that exponent. So, 4^-1 translates to 1/4, and 4^-2 becomes 1/(4*4), or 1/16. This principle helps in understanding that exponential functions can produce rapid increases with positive exponents, or rapid decreases—in the form of fractions—as the exponents become negative.

Exponents and Powers

Exponents and powers are foundational concepts in mathematics that play a crucial role in defining exponential functions. An exponent, often written as a small number above and to the right of a base number, signifies how many times the base is multiplied by itself. For instance, in 4^3 (read as 'four to the third power' or 'four cubed'), 4 is the base, and 3 is the exponent, indicating that 4 should be multiplied by itself twice (once for the second power, once for the third), resulting in 4*4*4, which equals 64.

Powers can also be zero or negative. When the exponent is zero, the power is always 1, reflecting the mathematical rule that any number to the zero power (except zero itself) equals one. In contrast, a negative exponent implies the reciprocal function. Therefore, recognizing how to work with exponents and powers is essential for accurately calculating and understanding the values of exponential functions.