Which of the following represents the graph of y=f(x) first shifted down 1 and reflected in the x-axis?
a) y=f(-x)-1
b)y=-(f(x)-1)
C)y=-f(x)-1
d) y=abs(f(x-1))
To understand how to shift a function down and reflect it in the x-axis, you need to know two basic transformations:
1. Vertical shifts: When a function is shifted up or down, the graph moves vertically in relation to the original position. A negative shift means the graph moves downward.
2. Reflection in the x-axis: When a function is reflected in the x-axis, every point on the graph switches its y-coordinate sign. For example, if a point is initially (x, y), the reflected point would be (x, -y).
Now let's analyze each option:
a) y=f(-x)-1:
This expression indicates that the graph is shifted horizontally and then shifted down. The "-x" inside the function implies a reflection in the y-axis, not the x-axis.
b) y=-(f(x)-1):
This expression reflects the graph vertically in the x-axis by negating the y-values, and then shifts it up by adding 1.
C) y=-f(x)-1:
This expression reflects the graph vertically in the x-axis by negating the y-values, and then shifts it down by subtracting 1. Therefore, this option represents the correct transformation.
d) y=abs(f(x-1)):
This option includes the absolute value function (|f(x-1)|), which doesn't represent a reflection in the x-axis. Instead, it represents a transformation that computes the absolute value of the function. So, this is not the correct answer.
In conclusion, the correct answer is and the graph that represents y=f(x) shifted down 1 and reflected in the x-axis is:
C) y=-f(x)-1.