To determine the equation of the quadratic function \( p(x) \) from the given table, we need to verify which one of the given vertex form equations accurately represents the values in the table.
The vertex form of a quadratic function is generally written as:
[tex]\[ p(x) = a(x - h)^2 + k \][/tex]
where \((h, k)\) is the vertex of the parabola.
Given the options:
1. \( p(x) = 2(x - 3)^2 - 1 \)
2. \( p(x) = 2(x + 3)^2 - 1 \)
3. \( p(x) = 3(x - 3)^2 - 1 \)
4. \( p(x) = 3(x + 3)^2 - 1 \)
We will go step-by-step through the possible equations to check which one fits the table values.
### Option 1: \( p(x) = 2(x - 3)^2 - 1 \)
We substitute the \( x \) values from the table into this equation and see if we get the corresponding \( p(x) \) values:
- For \( x = -1 \):
[tex]\[ p(-1) = 2(-1 - 3)^2 - 1 = 2(-4)^2 - 1 = 2 \cdot 16 - 1 = 32 - 1 = 31 \][/tex]
- For \( x = 0 \):
[tex]\[ p(0) = 2(0 - 3)^2 - 1 = 2(-3)^2 - 1 = 2 \cdot 9 - 1 = 18 - 1 = 17 \][/tex]
- For \( x = 1 \):
[tex]\[ p(1) = 2(1 - 3)^2 - 1 = 2(-2)^2 - 1 = 2 \cdot 4 - 1 = 8 - 1 = 7 \][/tex]
- For \( x = 2 \):
[tex]\[ p(2) = 2(2 - 3)^2 - 1 = 2(-1)^2 - 1 = 2 \cdot 1 - 1 = 2 - 1 = 1 \][/tex]
- For \( x = 3 \):
[tex]\[ p(3) = 2(3 - 3)^2 - 1 = 2(0)^2 - 1 = 2 \cdot 0 - 1 = 0 - 1 = -1 \][/tex]
- For \( x = 4 \):
[tex]\[ p(4) = 2(4 - 3)^2 - 1 = 2(1)^2 - 1 = 2 \cdot 1 - 1 = 2 - 1 = 1 \][/tex]
- For \( x = 5 \):
[tex]\[ p(5) = 2(5 - 3)^2 - 1 = 2(2)^2 - 1 = 2 \cdot 4 - 1 = 8 - 1 = 7 \][/tex]
All the values match the table exactly. Therefore, the correct quadratic function is:
[tex]\[ p(x) = 2(x - 3)^2 - 1 \][/tex]
This is option 1.
