A quadratic function y = f(x) is plotted on a graph and the vertex of the resulting parabola is (-4, 6,). What is the vertex of the function defined as g(x) = f(x+2) -2

Answer 1

When "x" is grouped with a number, such as "x + 2" the result is a horizontal translation of the function BUT the translation occurs in the opposite way as it is read. So the function f(x) is translated LEFT by 2 for f(x + 2). So the vertex of g(x) would have an x-value of -6.

When a number is added or subtracted outside the function rule (such as in this case "-2" it results in a vertical translation as it reads, so, f(x) is moved down by 2 to get g(x). That means the y-value of the vertex is 4.

So the vertex of (-4, 6) in f(x) is now (-6, 4) for g(x)

Answer 2

The graph of g(x) = f(x+2) - 2 is the parabola of f(x) but shifted 2 units to the LEFT (indicated by f(x+2)) and 2 units DOWN (indicated by -2).

Yes, the x translations are the opposite of what you would think: f(x+2) shifts the graph and vertex 2 units LEFT, not RIGHT. For illustration, y = x² has a vertex at x = 0, but y = (x+2)² has a vertex at x = -2 i.e. the vertex shifted 2 units LEFT, not RIGHT.

To obtain the location of the vertex of g(x), you have to subtract 2 units from the vertex x coordinate (shifting LEFT) and subtract 2 units from the vertex y coordinate (shifting DOWN):

Therefore, the vertex of g(x) will be at ( -4 - 2, 6 - 2) = ( -6, 4)