To find the appropriate input value that will result in an output of 20 and complete the table, let's analyze the given pairs and identify any pattern or function that applies to these pairs.
Given pairs:
- (0, 1)
- (1, 4)
- (2, 9)
- (5, 11)
- (6, 15)
We need to discover the function or pattern relating the input to the output and apply it to find the missing input for the output of 20.
### Step-by-Step Analysis:
1. Identify the nature of the relationship:
We observe the outputs increase with the inputs, usually a characteristic behavior of a polynomial (like [tex]\(y = ax^2 + bx + c\)[/tex]). However, the linear pattern also includes specific deviations (offsets).
2. Check if offsets exist:
Let's calculate the offset for each given input-output pair:
- For input 0, output 1: [tex]\( 0^2 + 1 = 1 \)[/tex]
- For input 1, output 4: [tex]\( 1^2 + 3 = 4 \)[/tex]
- For input 2, output 9: [tex]\( 2^2 + 5 = 9 \)[/tex]
- For input 5, output 11: [tex]\( 5^2 - 14 = 11 \)[/tex]
- For input 6, output 15: [tex]\( 6^2 - 21 = 15 \)[/tex]
We have identified the offsets:
- Offset for input 0 is 1.
- Offset for input 1 is 3.
- Offset for input 2 is 5.
- Offset for input 5 is -14.
- Offset for input 6 is -21.
3. Verify the consistency of the offsets and the relation:
The offsets seem to follow a pattern, but identifying a direct relation isn't immediately apparent. However, the pattern of offsets indicates the relationship may involve an increasing square term with varied offsets.
4. Find the missing input for the output 20:
Given the observed behavior, we find that the assumed relation follows [tex]\( y = x^2 + C \)[/tex], where C is an offset.
- Given output [tex]\( y = 20 \)[/tex], we aim to find the input [tex]\( x \)[/tex].
- We can note the previous offsets and infer the input for which this value holds. By calculating it:
For output 20: Presume offset behavior continues, then,
[tex]\( 20 = x^2 + C \)[/tex]
and our pattern suggests x^2 + a possible newly computed C based on our work above.
Hence, solving,
[tex]\( C = 20 - (7^2) = 20 - 49 = -29 \)[/tex] suggests an input closer in pattern at provided pairs suggests,
If all assignments align correctly then [tex]\( x = 7 \)[/tex], confirming our derived answer above aligns:
Thus,
the missing row in the table should be completed using:
Input 7 (to match derived pattern).
| Input | Output |
|-------|--------|
| 0 | 1 |
| 1 | 4 |
| 2 | 9 |
| 5 | 11 |
| 6 | 15 |
| \multicolumn{1}{|l|}{7} | 20 |
So, the completed final row for the table is:
Input: 7
