To determine the correct equation for [tex]\( p(x) \)[/tex] that fits the given data points, we'll analyze the options provided.
The data points given are:
- [tex]\( (-4, 24) \)[/tex]
- [tex]\( (-3, 9) \)[/tex]
- [tex]\( (-2, 0) \)[/tex]
- [tex]\( (-1, -3) \)[/tex]
- [tex]\( (0, 0) \)[/tex]
- [tex]\( (1, 9) \)[/tex]
- [tex]\( (2, 24) \)[/tex]
The possible equations of [tex]\( p(x) \)[/tex] given in vertex form are:
1. [tex]\( p(x) = 2(x - 1)^2 - 3 \)[/tex]
2. [tex]\( p(x) = 2(x + 1)^2 - 3 \)[/tex]
3. [tex]\( p(x) = 3(x - 1)^2 - 3 \)[/tex]
4. [tex]\( p(x) = 3(x + 1)^2 - 3 \)[/tex]
Let's identify the correct form of [tex]\( p(x) \)[/tex] by ensuring it fits all the data points. The equation verified correctly for each point is:
### Option 4: [tex]\( p(x) = 3(x + 1)^2 - 3 \)[/tex]
1. For [tex]\( x = -4 \)[/tex]:
[tex]\[ p(-4) = 3(-4 + 1)^2 - 3 \][/tex]
[tex]\[ p(-4) = 3(-3)^2 - 3 \][/tex]
[tex]\[ p(-4) = 3(9) - 3 \][/tex]
[tex]\[ p(-4) = 27 - 3 = 24 \][/tex]
2. For [tex]\( x = -3 \)[/tex]:
[tex]\[ p(-3) = 3(-3 + 1)^2 - 3 \][/tex]
[tex]\[ p(-3) = 3(-2)^2 - 3 \][/tex]
[tex]\[ p(-3) = 3(4) - 3 \][/tex]
[tex]\[ p(-3) = 12 - 3 = 9 \][/tex]
3. For [tex]\( x = -2 \)[/tex]:
[tex]\[ p(-2) = 3(-2 + 1)^2 - 3 \][/tex]
[tex]\[ p(-2) = 3(-1)^2 - 3 \][/tex]
[tex]\[ p(-2) = 3(1) - 3 \][/tex]
[tex]\[ p(-2) = 3 - 3 = 0 \][/tex]
4. For [tex]\( x = -1 \)[/tex]:
[tex]\[ p(-1) = 3(-1 + 1)^2 - 3 \][/tex]
[tex]\[ p(-1) = 3(0)^2 - 3 \][/tex]
[tex]\[ p(-1) = 3(0) - 3 \][/tex]
[tex]\[ p(-1) = -3 \][/tex]
5. For [tex]\( x = 0 \)[/tex]:
[tex]\[ p(0) = 3(0 + 1)^2 - 3 \][/tex]
[tex]\[ p(0) = 3(1)^2 - 3 \][/tex]
[tex]\[ p(0) = 3(1) - 3 \][/tex]
[tex]\[ p(0) = 3 - 3 = 0 \][/tex]
6. For [tex]\( x = 1 \)[/tex]:
[tex]\[ p(1) = 3(1 + 1)^2 - 3 \][/tex]
[tex]\[ p(1) = 3(2)^2 - 3 \][/tex]
[tex]\[ p(1) = 3(4) - 3 \][/tex]
[tex]\[ p(1) = 12 - 3 = 9 \][/tex]
7. For [tex]\( x = 2 \)[/tex]:
[tex]\[ p(2) = 3(2 + 1)^2 - 3 \][/tex]
[tex]\[ p(2) = 3(3)^2 - 3 \][/tex]
[tex]\[ p(2) = 3(9) - 3 \][/tex]
[tex]\[ p(2) = 27 - 3 = 24 \][/tex]
All the data points are satisfied by the equation [tex]\( p(x) = 3(x + 1)^2 - 3 \)[/tex].
Thus, the correct equation of [tex]\( p(x) \)[/tex] in vertex form is:
[tex]\[ \boxed{p(x) = 3(x + 1)^2 - 3} \][/tex]
