Function Table Completion: F(x) = 3x^2 + 3

Let's dive into the world of functions and complete this function table together! We'll be working with the function f(x) = 3x² + 3, which is a quadratic function. Don't worry if that sounds intimidating – we'll break it down step by step. This article will guide you through the process of evaluating the function for different values of x and filling in the missing pieces of the table. Understanding how functions work is a fundamental concept in mathematics, and this exercise will help you solidify your grasp on the topic. Function tables are a great way to visualize the relationship between input (x) and output (f(x)), making it easier to understand the behavior of the function. So, let's grab our mathematical tools and get started!

Understanding Function Tables

Before we jump into the calculations, let's quickly review what a function table is and why it's useful. A function table is a visual representation that organizes the input values (x) and their corresponding output values (f(x)) for a given function. Each row in the table represents a pair of x and f(x) values. By examining the table, we can easily see how the function transforms different input values into output values. This can help us identify patterns, understand the function's behavior, and even graph the function on a coordinate plane. Function tables are particularly helpful when dealing with different types of functions, including linear, quadratic, and exponential functions. They provide a clear and concise way to understand the relationship between variables and visualize the function's transformation. In essence, they serve as a powerful tool for understanding and analyzing functions in mathematics.

The Function: f(x) = 3x² + 3

The function we're working with today is f(x) = 3x² + 3. This is a quadratic function because the highest power of x is 2. Quadratic functions have a characteristic U-shape when graphed, called a parabola. The function takes an input x, squares it, multiplies the result by 3, and then adds 3. Let's break down each part:

• x²: This means we're squaring the input value x. For example, if x is 2, then x² is 2 * 2 = 4.
• 3x²: This means we're multiplying the squared value by 3. So, if x² is 4, then 3x² is 3 * 4 = 12.
• 3x² + 3: Finally, we add 3 to the result. If 3x² is 12, then 3x² + 3 is 12 + 3 = 15.

So, for an input of x = 2, the output f(x) would be 15. We'll use this process to fill in the missing values in our table. Remember that the order of operations (PEMDAS/BODMAS) is crucial here: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

The Table to Complete

Here's the table we need to complete:

Our mission is to find the corresponding f(x) values for each given x value. We'll do this by substituting each x value into the function f(x) = 3x² + 3 and calculating the result. Each calculation will fill in one of the blanks in the table. This hands-on approach will not only help us complete the table but also deepen our understanding of how the function behaves with different inputs. By the end of this process, we'll have a clear picture of the relationship between x and f(x) for this particular function.

Calculating f(x) for Each x Value

Now, let's calculate the f(x) values for each x value in the table. We'll go through each x value one by one, showing the steps involved in the calculation.

1. x = -1

To find f(-1), we substitute -1 for x in the function:

• f(-1) = 3(-1)² + 3
• First, we square -1: (-1)² = (-1) * (-1) = 1
• Then, we multiply by 3: 3 * 1 = 3
• Finally, we add 3: 3 + 3 = 6

So, f(-1) = 6. This means when x is -1, f(x) is 6.

2. x = 0

Next, let's find f(0) by substituting 0 for x:

• f(0) = 3(0)² + 3
• Squaring 0 gives us 0: (0)² = 0
• Multiplying by 3 still gives us 0: 3 * 0 = 0
• Adding 3 gives us the final result: 0 + 3 = 3

Therefore, f(0) = 3. When x is 0, f(x) is 3.

3. x = 1

Now, let's calculate f(1) by substituting 1 for x:

• f(1) = 3(1)² + 3
• Squaring 1 gives us 1: (1)² = 1
• Multiplying by 3 gives us 3: 3 * 1 = 3
• Adding 3 gives us the final result: 3 + 3 = 6

So, f(1) = 6. When x is 1, f(x) is 6.

4. x = 2

Finally, let's find f(2) by substituting 2 for x:

• f(2) = 3(2)² + 3
• Squaring 2 gives us 4: (2)² = 2 * 2 = 4
• Multiplying by 3 gives us 12: 3 * 4 = 12
• Adding 3 gives us the final result: 12 + 3 = 15

Therefore, f(2) = 15. When x is 2, f(x) is 15.

The Completed Table

Now that we've calculated all the f(x) values, let's fill in the table:

We have successfully completed the function table for f(x) = 3x² + 3! This table clearly shows how the output f(x) changes as the input x varies. We can observe that the function produces a symmetrical pattern around x = 0, which is characteristic of quadratic functions. The values increase as we move away from x = 0 in either direction. This is because the x² term dominates the function's behavior as x gets larger (either positive or negative). The constant term +3 simply shifts the entire graph upwards by 3 units.

Conclusion

Congratulations! You've successfully completed the function table for f(x) = 3x² + 3. You've learned how to evaluate a function for different input values and how to organize the results in a table. This is a valuable skill in mathematics and will help you understand more complex concepts in the future. Understanding functions is crucial for various mathematical applications, from graphing and data analysis to calculus and beyond. Function tables are a foundational tool that allows us to visualize and analyze the behavior of functions in a clear and organized manner. By practicing these types of exercises, you're building a strong foundation in mathematical thinking and problem-solving. Keep up the great work, and don't hesitate to explore more functions and tables!

For further exploration of functions, you can visit Khan Academy's Functions and equations section.