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))
why and how do you know
Which of the following equations represents the graph of
𝑓(𝑥)=𝑥+1
f
(
x
)
=
x
+
1
after stretching vertically by the factor of 2?
To answer this question, we need to understand how each operation affects the original function.
First, let's consider the original function, y = f(x).
1. Shifting down 1: To shift a function down by a certain value, we subtract that value from the function. Therefore, y = f(x) shifted down 1 would be y = f(x) - 1.
2. Reflecting in the x-axis: To reflect a function in the x-axis, we take the negative of the function's y-values. So if the original function is y = f(x), the reflected function in the x-axis would be y = -f(x).
Now let's analyze the answer choices:
a) y = f(-x) - 1: This answer choice represents the original function being reflected horizontally (by replacing x with -x) but does not include the reflection in the x-axis. Therefore, it is not the correct answer.
b) y = -(f(x) - 1): This answer choice includes both the reflection in the x-axis (taking the negative of the function f(x)) and the shift down 1 (subtracting 1 from the function). So, this represents the graph of y = f(x) first shifted down 1 and then reflected in the x-axis. We can consider this answer as a possibility.
c) y = -f(x) - 1: This answer choice includes the reflection in the x-axis but does not include the shift down 1. Therefore, it does not represent the correct transformation.
d) y = abs(f(x-1)): This answer choice represents the absolute value of the original function with a horizontal shift of 1 unit to the right. It does not involve the reflection in the x-axis or the shift down 1, so it does not represent the correct transformation.
In conclusion, the correct answer is b) y = -(f(x) - 1).