To determine which equation matches the given points, we will check each quadratic function provided against the given data points [tex]\((-5, -23), (-4, -13), (-3, -7), (-2, -5), (-1, -7), (0, -13), (1, -23)\)[/tex].
Consider the four possible equations:
1. [tex]\( y = -2(x + 2)^2 - 5 \)[/tex]
2. [tex]\( y = -2(x - 2)^2 + 5 \)[/tex]
3. [tex]\( y = -(x + 2)^2 - 5 \)[/tex]
4. [tex]\( y = -(x - 2)^2 + 5 \)[/tex]
Let's substitute the [tex]\( x \)[/tex] values from the given points into each equation to see which one produces the corresponding [tex]\( y \)[/tex] values.
### Testing Equation 1: [tex]\( y = -2(x + 2)^2 - 5 \)[/tex]
- For [tex]\( x = -5 \)[/tex]:
[tex]\[
y = -2(-5 + 2)^2 - 5 = -2(-3)^2 - 5 = -2 \cdot 9 - 5 = -18 - 5 = -23
\][/tex]
This matches the point [tex]\((-5, -23)\)[/tex].
- For [tex]\( x = -4 \)[/tex]:
[tex]\[
y = -2(-4 + 2)^2 - 5 = -2(-2)^2 - 5 = -2 \cdot 4 - 5 = -8 - 5 = -13
\][/tex]
This matches the point [tex]\((-4, -13)\)[/tex].
- For [tex]\( x = -3 \)[/tex]:
[tex]\[
y = -2(-3 + 2)^2 - 5 = -2(-1)^2 - 5 = -2 \cdot 1 - 5 = -2 - 5 = -7
\][/tex]
This matches the point [tex]\((-3, -7)\)[/tex].
- For [tex]\( x = -2 \)[/tex]:
[tex]\[
y = -2(-2 + 2)^2 - 5 = -2(0)^2 - 5 = 0 - 5 = -5
\][/tex]
This matches the point [tex]\((-2, -5)\)[/tex].
- For [tex]\( x = -1 \)[/tex]:
[tex]\[
y = -2(-1 + 2)^2 - 5 = -2(1)^2 - 5 = -2 \cdot 1 - 5 = -2 - 5 = -7
\][/tex]
This matches the point [tex]\((-1, -7)\)[/tex].
- For [tex]\( x = 0 \)[/tex]:
[tex]\[
y = -2(0 + 2)^2 - 5 = -2(2)^2 - 5 = -2 \cdot 4 - 5 = -8 - 5 = -13
\][/tex]
This matches the point [tex]\((0, -13)\)[/tex].
- For [tex]\( x = 1 \)[/tex]:
[tex]\[
y = -2(1 + 2)^2 - 5 = -2(3)^2 - 5 = -2 \cdot 9 - 5 = -18 - 5 = -23
\][/tex]
This matches the point [tex]\((1, -23)\)[/tex].
Since the equation [tex]\( y = -2(x + 2)^2 - 5 \)[/tex] matches all the given points, it is the correct equation for the function.
### Conclusion
The correct equation that matches the given data points is:
[tex]\[ y = -2(x + 2)^2 - 5 \][/tex]
