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\begin{tabular}{|l|l|}
\hline -4 & 24 \\
\hline -3 & 9 \\
\hline -2 & 0 \\
\hline -1 & -3 \\
\hline 0 & 0 \\
\hline 1 & 9 \\
\hline 2 & 24 \\
\hline
\end{tabular}

What is the equation of [tex]$p(x)$[/tex] in vertex form?

A. [tex]$p(x) = 2(x-1)^2 - 3$[/tex]

B. [tex]$p(x) = 2(x+1)^2 - 3$[/tex]

C. [tex]$p(x) = 3(x-1)^2 - 3$[/tex]

D. [tex]$p(x) = 3(x+1)^2 - 3$[/tex]

Sagot :

GinnyAnsweridn
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]
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