Given the data points:
| [tex]\( x \)[/tex] | [tex]\( y \)[/tex] |
|---------|-------------|
| -8 | -576.64 |
| -7 | -441.49 |
| -6 | -324.36 |
| -5 | -225.25 |
| -4 | -144.16 |
We are looking for a function in the form [tex]\( y = a x^2 \)[/tex] that fits this data.
To determine the value of [tex]\( a \)[/tex] in the quadratic model:
1. Take any data point from the table. For example, let's use the data point [tex]\((-8, -576.64)\)[/tex].
2. Since we assume the model [tex]\( y = a x^2 \)[/tex], substitute [tex]\( x = -8 \)[/tex] and [tex]\( y = -576.64 \)[/tex] into the equation:
[tex]\[ -576.64 = a \cdot (-8)^2 \][/tex]
3. Simplify the equation by calculating [tex]\((-8)^2 = 64\)[/tex]:
[tex]\[ -576.64 = a \cdot 64 \][/tex]
4. Solve for [tex]\( a \)[/tex]:
[tex]\[ a = \frac{-576.64}{64} \][/tex]
[tex]\[ a \approx -9.01 \][/tex]
Therefore, the quadratic function that models the given data is:
[tex]\[ y = -9.01 x^2 \][/tex]
