Question

Which equation represents the graph of an absolute value function that has been horizontally stretched by a factor of 2 and shifted up 3 units?

A) y = 2| x | + 3
B) y = | 2x | + 3
C) y = | x | + 2
D) y = | x | + 3
E) y = | x/2 | + 3

Answer 1

The equation representing an absolute value function stretched horizontally by a factor of 2 and shifted up by 3 units is E) y = | x/2 | + 3. This accounts for the reciprocal stretch factor inside the absolute value and the vertical shift in the function.

Step-by-step explanation:

The question asks which equation represents an absolute value function that has been horizontally stretched by a factor of 2 and shifted up 3 units. The general form of an absolute value function is y = a|b(x - h)| + k, where a is the vertical stretch or compression, b is the horizontal stretch or compression (which is the reciprocal of the actual stretch factor), h is the horizontal shift, and k is the vertical shift.

For a horizontal stretch by a factor of 2, we would divide x by 2 inside the absolute value, resulting in b = 1/2. A shift up by 3 units is represented by k = +3. Therefore, the correct equation is y = |x/2| + 3, since this indicates a horizontal stretch by a factor of 2 (due to the reciprocal in the b term) and a vertical shift upwards by 3 units. The correct answer is E) y = | x/2 | + 3.