Consider the function f(x) = x^2 -5. If g(x) = f(x-7), what can be said of g(x)? Check all that apply from the following. Show your work.
a) The graph of g(x) is shifted seven units to the left of f(x)
b) The graph of g(x) is shifted seven units to the right of f(x)
c) The graph of g(x) is shifted seven units above f(x)
d) The graph of g(x) is shifted seven units below f(x)
e) The vertex of g(x) is (7, 0)
f) The axis of symmetry of g(x) is x = 7.
To find g(x), we need to substitute x-7 into f(x). So g(x) = (x-7)^2 - 5 = x^2 - 14x + 49 - 5 = x^2 - 14x + 44.
Now, let's analyze the properties of g(x) compared to f(x):
a) The graph of g(x) is shifted seven units to the left of f(x):
This is correct because the vertex of f(x) is at (0, -5) while the vertex of g(x) is at (7, 0).
b) The graph of g(x) is shifted seven units to the right of f(x)
This is incorrect.
c) The graph of g(x) is shifted seven units above f(x):
This is incorrect because the vertex of g(x) is actually below the vertex of f(x.
d) The graph of g(x) is shifted seven units below f(x)
This is correct because the vertex of g(x) is at (7, 0) which is below the vertex of f(x) at (0, -5).
e) The vertex of g(x) is (7, 0)
This is correct as shown in the calculations.
f) The axis of symmetry of g(x) is x = 7
This is correct because the axis of symmetry of a parabola is given by x = -b/(2a) and for g(x) we have a = 1, b = -14, so x = 7.