By analyzing first and second differences, the missing y-values for a polynomial function of degree 2 with x-values -3, -2, -1, 0, 1, 2, 3 were determined. The filled-in y-values are 1, -5, -7, -9, maintaining a constant second difference of 2.
To find the missing y-values and complete the table, we can use the given first and second differences. The second differences being constant (2) indicates that the polynomial function is of degree 2.
Given:
\[ x = -3, -2, -1, 0, 1, 2, 3 \]
\[ y = 15, 7, ?, -3, ?, -5, ? \]
\[ \text{1st diff} = -8, ?, -4, -2, 0, ? \]
\[ \text{2nd diff} = 2, 2, 2, 2, 2 \]
Now, let's fill in the missing y-values using the differences:
1. Starting from the given y-value (15) and subtracting the corresponding first differences:
\[ 15 - (-8) = 23 \] (for the missing y-value after 15)
2. For the next missing y-value, subtracting the corresponding first difference:
\[ 7 - (-8) = 15 \]
3. For the next missing y-value:
\[ -3 - (-4) = -3 + 4 = 1 \]
4. For the next missing y-value:
\[ -5 - 0 = -5 \]
Now, the completed table is:
\[ x = -3, -2, -1, 0, 1, 2, 3 \]
\[ y = 15, 7, 1, -3, -5, -7, -9 \]
\[ \text{1st diff} = -8, -6, -4, -2, 0, 2, 4 \]
\[ \text{2nd diff} = 2, 2, 2, 2, 2, 2, 2 \]
The filled-in y-values maintain the constant second differences, confirming the polynomial function is of degree 2.