Question

Review the graphs of f(x) = –x5 + 2x4 + 1 and g(x), which is a translation of f(x).

On a coordinate plane, g (x) curves down through (negative 3, 3) to (negative 2, 1) and then curves up to (negative 0.5, 3.8). It then curves down through (0, 0) into quadrant 4. y = f (x) curves down through (negative 1, 3) to (0, 1) and then curves up to (1.5, 3.8). It then curves down through (2, 0).

How can g(x) be expressed in terms of f(x)?

g(x) = (–x5 + 2x4 + 1) + 2
g(x) = –(x + 2)5 + 2(x + 2)4 + 1
g(x) = (–x5 + 2x4 + 1) – 2
g(x) = –(x – 2)5 + 2(x – 2)4 + 1

Answer 1

g(x) is a translation of f(x) and can be correctly expressed as g(x) = −(x + 2)^5 + 2(x + 2)^4 + 1, revealing a shift two units to the left of f(x).

Step-by-step explanation:

To determine how g(x) can be expressed in terms of f(x), we need to identify the type of transformation that has occurred from f(x) = −x5 + 2x4 + 1 to g(x). Given the description of g(x), it is clear that this translation involves a shift of the original function f(x).

First, we will examine the given points through which y = f(x) passes, namely (−1, 3), (0, 1), and (1.5, 3.8). Then we compare with the corresponding points given for g(x): (−3, 3), (−2, 1), and (−0.5, 3.8). By examining these points, we can see that g(x) is shifted two units to the left of f(x) since each x-value for g(x) is less by 2 compared to the corresponding x-value for f(x).

Therefore, the translation transformation is f(x + 2), which shifts the graph of f(x) two units to the left.

Thus, g(x) can be correctly expressed as g(x) = −(x + 2)5 + 2(x + 2)4 + 1, which is a translation of two units to the left.

Answer 2

choice B

g(x) = –(x + 2)5 + 2(x + 2)4 + 1