Let's determine the equation of [tex]\( h(x) \)[/tex] in vertex form using the given points in the table. The vertex form of a quadratic function is given by:
[tex]\[ h(x) = a(x - h)^2 + k \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
We are provided with four possible equations:
1. [tex]\( h(x) = (x - 2)^2 + 3 \)[/tex]
2. [tex]\( h(x) = (x - 1)^2 + 2 \)[/tex]
3. [tex]\( h(x) = (x + 1)^2 + 2 \)[/tex]
4. [tex]\( h(x) = (x + 2)^2 + 3 \)[/tex]
To find the correct equation, we need to check which of these equations fits all the given points.
Let's analyze the points one by one for each equation.
First, check [tex]\( h(x) \)[/tex] for [tex]\( h(x) = (x - 2)^2 + 3 \)[/tex]:
1. For [tex]\( x = -3 \)[/tex], [tex]\( h(-3) = ((-3) - 2)^2 + 3 = 25 + 3 = 28 \)[/tex] (does not match 6)
2. This fails, so option 1 is incorrect.
Second, check [tex]\( h(x) \)[/tex] for [tex]\( h(x) = (x - 1)^2 + 2 \)[/tex]:
1. For [tex]\( x = -3 \)[/tex], [tex]\( h(-3) = ((-3) - 1)^2 + 2 = 16 + 2 = 18 \)[/tex] (does not match 6)
2. This fails, so option 2 is incorrect.
Third, check [tex]\( h(x) \)[/tex] for [tex]\( h(x) = (x + 1)^2 + 2 \)[/tex]:
1. For [tex]\( x = -3 \)[/tex], [tex]\( h(-3) = ((-3) + 1)^2 + 2 = 4 + 2 = 6 \)[/tex] (matches 6)
2. For [tex]\( x = -2 \)[/tex], [tex]\( h(-2) = ((-2) + 1)^2 + 2 = 1 + 2 = 3 \)[/tex] (matches 3)
3. For [tex]\( x = -1 \)[/tex], [tex]\( h(-1) = ((-1) + 1)^2 + 2 = 0 + 2 = 2 \)[/tex] (matches 2)
4. For [tex]\( x = 0 \)[/tex], [tex]\( h(0) = (0 + 1)^2 + 2 = 1 + 2 = 3 \)[/tex] (matches 3)
5. For [tex]\( x = 1 \)[/tex], [tex]\( h(1) = (1 + 1)^2 + 2 = 4 + 2 = 6 \)[/tex] (matches 6)
6. For [tex]\( x = 2 \)[/tex], [tex]\( h(2) = (2 + 1)^2 + 2 = 9 + 2 = 11 \)[/tex] (matches 11)
7. For [tex]\( x = 3 \)[/tex], [tex]\( h(3) = (3 + 1)^2 + 2 = 16 + 2 = 18 \)[/tex] (matches 18)
Thus, all points match for [tex]\( h(x) = (x + 1)^2 + 2 \)[/tex].
Fourth, check [tex]\( h(x) \)[/tex] for [tex]\( h(x) = (x + 2)^2 + 3 \)[/tex]:
1. For [tex]\( x = -3 \)[/tex], [tex]\( h(-3) = ((-3) + 2)^2 + 3 = 1 + 3 = 4 \)[/tex] (does not match 6)
2. This fails, so option 4 is incorrect.
Conclusively, the correct equation of [tex]\( h(x) \)[/tex] in vertex form is:
[tex]\[ h(x) = (x + 1)^2 + 2 \][/tex]
Which corresponds to the third option.
