Question

Find a simple function that fits the data in the tables. $$\begin{array}{|r|r|} \hline x & y \\ \hline 0 & -1 \\ \hline 1 & 0 \\ \hline 4 & 1 \\ \hline 9 & 2 \\ \hline 16 & 3 \\ \hline \end{array}$$

Short answer

Question: Determine the simple function that fits the given data in the table below. | x | y | |-----|-----| | 0 | -1 | | 1 | 0 | | 4 | 1 | | 9 | 2 | | 16 | 3 | Answer: The simple function that fits the data is $$y(x) = \sqrt{x} - 1$$.

Step-by-step solution

Identify the pattern

Start by examining the x and y values in the table. Notice that for each value of x, x is a perfect square, meaning x is the square of some integer. Now, look at the corresponding y value and observe that y is one less than the square root of x. The pattern can be described as: $$ y = \sqrt{x} - 1 $$

Test the found pattern

To ensure that the found pattern holds true for all values in the table, we can test it by plugging in each value of x and checking if it results in the corresponding y value: $$ \begin{aligned} y(0) &= \sqrt{0} - 1=-1 \\ y(1) &= \sqrt{1} - 1 = 0 \\ y(4) &= \sqrt{4} - 1 = 1 \\ y(9) &= \sqrt{9} - 1 = 2 \\ y(16) &= \sqrt{16} - 1 = 3 \end{aligned} $$

Present the function

Since the pattern we found works for all values of x in the table, we can now present our simple function that fits the data: $$ y(x) = \sqrt{x} - 1 $$

Key concepts

Pattern Identification

When working with mathematical data, identifying patterns is a crucial step. Imagine the x and y values in the table like puzzle pieces where you’re trying to find a repeated design.
For this data:

• x-values are organized in a recognizable pattern where each x is a perfect square.
• The corresponding y-values appear to follow another pattern: they are consistently one less than the square root of x.

Recognizing these sequences helps in establishing a rule or formula that links x and y values. This systematic approach to observing data not only simplifies problems but also develops a solid foundation for function fitting, which is a technique that projects a mathematical formula that aligns with set points or data.

Perfect Squares

Perfect squares play an important role in this kind of exercise. They are numbers which are squares of integers and have a distinct pattern:

• The sequence starts at 0 (since 0 squared is 0), and includes numbers like 1, 4, 9, 16, etc., which correspond to 1², 2², 3², and 4², respectively.

Understanding perfect squares can make spotting patterns intuitive and straightforward. In the exercise, x-values were these very perfect squares, acting as a setup for a simple equation where their square roots were used in deriving the y-values.
Recognizing perfect squares allows you to swiftly see relationships between numbers and offer insight for potential mathematical models or equations that can accurately portray the data in the table.

Square Root Function

The square root function pulls numbers toward their square roots, and is massively useful in various mathematical problems. In the context of this exercise:

• Y-values of the data are derived by taking the square root of the x-values, then subtracting 1. This essentially creates a square root function adjusted by a constant -1.
• The formula represents this relation as \( y = \sqrt{x} - 1 \).

This simple equation fits perfectly with the data, providing a clear illustration of the role of square root functions. By understanding these functions, you can model real-life situations or problems where relationships behave similarly, making them a powerful tool in both pure and applied mathematics.