Question

What is the equation of the graph below?

Answer 1

Looks like an inverted parabola with vertex at (-2,2). Do you know how to get the equation from that information?

Answer 2

Not sure how to

Answer 3

i know the choices are y = –4(x + 2)2 + 2

y = –4(x – 2)2 + 2

y = 4(x – 2)2 – 2

y = 4(x + 2)2 + 2

Answer 4

I know it's not C

Answer 5

Well, time to either refresh your memory, or reason it out. Which do you prefer? :-)

Reasoning it out is an excellent choice!

Okay, we know that this is an inverted parabola, so it must be a - sign in front of the x^2 portion. That eliminates the last two choices, which obviously open upward if you try a few points (x=2,3,4 gives y =-2,2,14 for the 3rd choice, and 66, 102,146 for the 4th)

Going back to theory for a moment, if you have a function y = f(x), replacing x with x-2 gives you y = f(x-2). That's the same as shifting the whole graph 2 units to the left — where you used to have some feature of the graph at x, it now appears at x-2, right?
Similarly, if we replaced x with x+2, the graph shifts to the right, and something that used to appear at x = 0 would now appear at x=2.

Also, there's the shift up / down to account for. Well, that's just adding or subtracting something from the result of f(x). y = f(x)+2 is just going to give us the same graph, shifted up 2 points. y = f(x)-2 gives us the same graph, shifted down two points.

Does that make sense? Knowing that, which equation is your choice?

Answer 6

i think it's B because negative shifts to the left, and the parabola is on the left side of the graph

Answer 7

We can classify all of the choices:
A: shifted right by 2, and up by 2, opens down
B: shifted left by 2, up by 2, opens down
C: shifted left by 2, down by 2, opens up
D: shifted right by 2, up by 2, opens up

Answer 8

so B is the correct answer.

we could also use vertex form for a parabola:

\[y = a(x-h)^2 + k\] where the vertex is at \((h,k)\)
\(a\) tells us whether it opens up or down, positive \(a\) opens up, otherwise it opens down.

Answer 9

I hope that helps, I have to leave now.

Answer 10

thank you very much for the thorough explanations!