the relationship between the graphs f(x) and -f(x) is:
a)a reflection about the line y=x
b)a reflection in y-axis
c)a reflection in x-axis
d) a reflection of origin
e) a reflection about the line y=-x
well, the y-value has changed sign, so that means what was above the x-axis is now below, right?
To understand the relationship between the graphs of f(x) and -f(x), let's break it down step-by-step.
The function -f(x) represents the opposite or negative of f(x). This means that the values of -f(x) are obtained by multiplying the values of f(x) by -1.
Now, let's consider how this affects the graph of f(x):
a) Reflection about the line y = x:
In this case, we would need to swap the x and y coordinates of each point on the graph. However, multiplying f(x) by -1 does not accomplish this transformation, so option (a) is not correct.
b) Reflection in the y-axis:
When we reflect a graph in the y-axis, we need to negate the x-coordinate of each point. Multiplying f(x) by -1 achieves exactly this transformation. So, the graph of -f(x) is indeed a reflection of f(x) in the y-axis. Therefore, option (b) is correct.
c) Reflection in the x-axis:
A reflection in the x-axis involves negating the y-coordinate of each point. Transforming f(x) to -f(x) only changes the sign of the y-values, but it does not reflect the graph over the x-axis. Therefore, option (c) is not correct.
d) Reflection of the origin:
To reflect a graph about the origin (0,0), we would need to negate both the x and y coordinates. Transforming f(x) to -f(x) only negates the y-values, so this transformation is not a reflection of the origin. Thus, option (d) is not correct.
e) Reflection about the line y = -x:
Similar to option (a), reflecting a graph about the line y = -x involves swapping the x and y coordinates of each point. Since multiplying f(x) by -1 only negates the y-values, it does not produce a reflection about the line y = -x. Therefore, option (e) is not correct.
In conclusion, the correct answer is b) a reflection in the y-axis.