To find the equation of [tex]\( h(x) \)[/tex] in vertex form using the given points, we have to test which equation fits all the data points in the table.
The possible equations are:
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]
We will verify which equation correctly calculates the values for all given [tex]\( x \)[/tex] in the table.
1. Test [tex]\( h(x) = (x - 2)^2 + 3 \)[/tex]:
- For [tex]\( x = -3 \)[/tex]: [tex]\( h(-3) = (-3 - 2)^2 + 3 = (-5)^2 + 3 = 25 + 3 = 28 \)[/tex] (not matching)
- So, this equation is incorrect.
2. Test [tex]\( h(x) = (x - 1)^2 + 2 \)[/tex]:
- For [tex]\( x = -3 \)[/tex]: [tex]\( h(-3) = (-3 - 1)^2 + 2 = (-4)^2 + 2 = 16 + 2 = 18 \)[/tex] (not matching)
- So, this equation is incorrect.
3. Test [tex]\( h(x) = (x + 1)^2 + 2 \)[/tex]:
- For [tex]\( x = -3 \)[/tex]: [tex]\( h(-3) = (-3 + 1)^2 + 2 = (-2)^2 + 2 = 4 + 2 = 6 \)[/tex] (matching)
- For [tex]\( x = -2 \)[/tex]: [tex]\( h(-2) = (-2 + 1)^2 + 2 = (-1)^2 + 2 = 1 + 2 = 3 \)[/tex] (matching)
- For [tex]\( x = -1 \)[/tex]: [tex]\( h(-1) = (-1 + 1)^2 + 2 = (0)^2 + 2 = 0 + 2 = 2 \)[/tex] (matching)
- For [tex]\( x = 0 \)[/tex]: [tex]\( h(0) = (0 + 1)^2 + 2 = (1)^2 + 2 = 1 + 2 = 3 \)[/tex] (matching)
- For [tex]\( x = 1 \)[/tex]: [tex]\( h(1) = (1 + 1)^2 + 2 = (2)^2 + 2 = 4 + 2 = 6 \)[/tex] (matching)
- For [tex]\( x = 2 \)[/tex]: [tex]\( h(2) = (2 + 1)^2 + 2 = (3)^2 + 2 = 9 + 2 = 11 \)[/tex] (matching)
- For [tex]\( x = 3 \)[/tex]: [tex]\( h(3) = (3 + 1)^2 + 2 = (4)^2 + 2 = 16 + 2 = 18 \)[/tex] (matching)
- All points match, so this equation is correct.
4. Test [tex]\( h(x) = (x + 2)^2 + 3 \)[/tex]:
- For [tex]\( x = -3 \)[/tex]: [tex]\( h(-3) = (-3 + 2)^2 + 3 = (-1)^2 + 3 = 1 + 3 = 4 \)[/tex] (not matching)
- So, this equation is incorrect.
Based on the evaluation, the correct equation of [tex]\( h(x) \)[/tex] in vertex form is:
[tex]\[ h(x) = (x + 1)^2 + 2 \][/tex]
So, the answer is [tex]\( h(x) = (x + 1)^2 + 2 \)[/tex], which corresponds to the third option, therefore:
[tex]\[ 3 \][/tex] is the correct choice.
