Question

The table represents a quadratic function.


x y
−6 23
−5 8
−4 −1
−3 −4
−2 −1
−1 8
0 23

What is the equation of the function?
y = (x + 3)2 − 4
y = (x − 3)2 + 4
y = 3(x + 3)2 − 4
y = 3(x − 3)2 + 4

PLEASE EXPLAIN YOUR ANSWER

Answer 1

You could try plugging in coordinates. For example, (0, 23) might easiest since x would just be 0.

Using the vertex formula (h, k) , where it's (-3, -4) and plugging in what you can to \[y=a(x-h)^2 +k\]
\[ y=a (x+3)^2-4 \]

You can narrow it down to either "A" or "C"
Just plug in (0, 23) and see which gives you a true statement.

Answer 2

let\[y=ax^2+bx+c ~...(1)\]
when x=0,y=23
23=0+0+c
c=23
\[y=ax^2+bx+23~...(2)\]
when x=-1,y=8
\[8=a(-1)^2+b(-1)+23\]
8=a-b+23
a-b=8-23=-15
a-b=-15 ...(3)
when x=-2,y=-1
\[-1=a(-2)^2+b(-2)+23 \]
-1=4a-2b+23
4a-2b=-1-23
4a-2b=-24
divide by 2
2a-b=-12 ...(4)
subtract (3)from (4)
a=-12+15
a=3
from (3)
3-b=-15
-b=-15-3
b=18
from (1)
\[y=3x^2+18x+23\]
\[[y=3(x^2+6x+9-9)+23\]
\[y=3(x^2+6x+9)-27+23\]
y=?