How to Find the Function Rule of a Table?

Finding the function rule from a table involves determining the relationship between the input (x) and the output (y) values. This relationship can often be expressed as an equation. The following steps outline how to determine the function rule, particularly if it's a linear function.

Steps to Determine the Function Rule

1. Identify the pattern: Look at the table and see how the 'y' values change as the 'x' values change. Is there a constant addition, subtraction, multiplication, or division?

2. Check for a Linear Function: The reference highlights checking if the table follows a linear function, which can be represented in the form y = ax + b, where a is the slope and b is the y-intercept.

3. Calculate the Slope (a): If you suspect a linear relationship, calculate the slope (a) using two points from the table (x₁, y₁) and (x₂, y₂):

   • a = (y₂ - y₁) / (x₂ - x₁)

4. Find the y-intercept (b): Once you have the slope (a), substitute the values of x and y from any point in the table into the equation y = ax + b and solve for b.

5. Write the Function Rule: Now that you have both a and b, you can write the function rule in the form y = ax + b.

Example

Let's say you have the following table:

x	y
0	1
1	3
2	5
3	7

1. Pattern: As x increases by 1, y increases by 2.

2. Linear Check: Let's assume it's linear and try to find a and b.

3. Slope (a): Using points (0, 1) and (1, 3):

   • a = (3 - 1) / (1 - 0) = 2 / 1 = 2

4. Y-intercept (b): Using the point (0, 1) and the slope a = 2:

   • 1 = 2(0) + b
   • 1 = 0 + b
   • b = 1

5. Function Rule: Therefore, the function rule is y = 2x + 1.

Using Equations from the Table

As the reference states, you can build a set of equations. For instance, using the same table:

• q(x) = ax + b
• q(0) = a(0) + b = 1 (from the point (0,1))
• q(1) = a(1) + b = 3 (from the point (1,3))

From the first equation, we get b = 1. Substituting b into the second equation:

• a + 1 = 3
• a = 2

This confirms our function rule is y = 2x + 1.

Important Considerations

• Not all tables represent functions that can be easily expressed with a simple equation. Some relationships might be more complex (quadratic, exponential, etc.) or not follow a clear mathematical pattern.
• Always check multiple points in the table to ensure the rule holds true. This increases the likelihood that you've found the correct function rule.
• For non-linear functions, more advanced techniques are required to find the function rule. This may involve recognizing patterns specific to quadratic, exponential, or other types of functions.