Problem 15 - Algebra 1: Concepts and Skills

Question

Make a table of values for the exponential function. Use \(x\) -values of \(-2,-1,0,1,2,\) and 3. $$y=3^{x}$$

Step-by-Step Solution

Answer

The table of values is: \(-2, 0.11\), \(-1, 0.33\), \(0, 1\), \(1, 3\), \(2, 9\), and \(3, 27\).

Step 1: Substitution for \(x = -2\)

First, substitute \(x = -2\) into the equation. This results in \(y = 3^{-2}\). Calculate this value.

Step 2: Substitution for \(x = -1\)

Next, substitute \(x = -1\) into the equation. This results in \(y = 3^{-1}\). Calculate this value.

Step 3: Substitution for \(x = 0\)

Substitute \(x = 0\) into the equation. This results in \(y = 3^{0}\). Calculate this value.

Step 4: Substitution for \(x = 1\)

Now, substitute \(x = 1\) into the equation. This results in \(y = 3^{1}\). Calculate this value.

Step 5: Substitution for \(x = 2\)

Substitute \(x = 2\) into the equation. This results in \(y = 3^{2}\). Calculate this value.

Step 6: Substitution for \(x = 3\)

Lastly, substitute \(x = 3\) into the equation. This results in \(y = 3^{3}\). Calculate this value.

Step 7: Create the Table

Using these results, create a table. Each row should contain the \(x\) value and the corresponding \(y\) value.

Key Concepts

Table of Values

Creating a table of values is a handy way to organize and analyze the behavior of an equation or function. In our example with the exponential function \(y = 3^x\), the table showcases how the dependent variable \(y\) changes as the independent variable \(x\) varies.

To form the table, you substitute different values of \(x\), such as -2, -1, 0, 1, 2, and 3, into the equation, then calculate the resulting \(y\).

• For \(x = -2\), \(y = 3^{-2}\)
• For \(x = -1\), \(y = 3^{-1}\)
• For \(x = 0\), \(y = 3^{0}\)
• For \(x = 1\), \(y = 3^{1}\)
• For \(x = 2\), \(y = 3^{2}\)
• For \(x = 3\), \(y = 3^{3}\)

Writing down each pair of numbers creates the table structure, helping to visualize how \(y\) increases or decreases in response to changes in \(x\). It's a straightforward method to understand the relationship within the function.

Substitution Method

The substitution method is a primary technique in algebra that helps to find specific values for an equation. Within the context of exponential functions such as \(y = 3^x\), substitution involves replacing the variable \(x\) with given numbers to evaluate \(y\).

Let's see how it works practically through these steps:

• Pick a value for \(x\), say \(-2\).
• Replace \(x\) in the equation: \(y = 3^{-2}\).
• Calculate the result, \(y = \frac{1}{9}\).

Repeat this method for other \(x\) values like \(-1, 0, 1, 2,\) and \(3\). Each substitution gives you the corresponding \(y\) value, like solving little puzzles. By doing so, you create a comprehensive table of values, vital for understanding function behavior.

Exponents

Exponents are the foundation of exponential functions like \(y = 3^x\). They represent the number of times a base (here, 3) is multiplied by itself. Understanding exponents is key to mastering these functions.

• \(3^{-2}\) means \(\frac{1}{3 \times 3}\), resulting in \(\frac{1}{9}\).
• \(3^0\) denotes any number raised to the power of zero, which is always 1.
• \(3^3\) represents \(3 \times 3 \times 3\), leading to 27.

Exponents can be positive, negative, or zero. Each type has special rules:

• Positive exponents imply repeated multiplication: \(3^2 = 9\).
• Zero exponent results in 1: \(3^0 = 1\).
• Negative exponents denote division: \(3^{-1} = \frac{1}{3}\).

Grasping these rules makes solving exponential functions much simpler, aiding in evaluating expressions and graphing these curves correctly.

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating those symbols, centered around solving equations. In our exercise, algebra helps us manage the expression \(y = 3^x\), and through steps of substitution and computation, we find values that define and solve the function's behavior.

• Substitution: Replacing variables with numbers to compute results.
• Function Evaluation: Using known rules to determine the outcome of expressions like \(3^x\).
• Simplification: Breaking down complex expressions, \(3^{-2} = \frac{1}{9}\).

By applying algebraic principles, you can dissect any exponential function, catalog data in a table, and reveal patterns or trends within the table. Whether for math class or real-world problems, algebra is a vital tool that simplifies the process of working with various types of equations and functions.