Which of the following results in the graph of f(x)=x^2 being expanded vertically by a factor of 3 and?
reflected over the x-axis?
a. f(x)=1/3x^2
b. f(x)= -3x^2
c. f(x)= -1/x^2 +3
d. f(x)= -1/3 x^2
perform your transformation on an arbitrary point.
e.g. (2,4) is on the original graph
so a vertical expansion by a factor of 3 would make it
(2,4) ----> (2,12)
then reflect that in the x-axis
(2,12) ----> (2,-12)
now which of your choices would yield that result?
To determine which of the given options results in the graph of f(x) = x^2 being expanded vertically by a factor of 3 and reflected over the x-axis, we need to identify the transformations that affect these changes.
The transformation that expands a function vertically by a factor of "a" is represented by multiplying the function by "a." In this case, since we want a vertical expansion by a factor of 3, the function needs to be multiplied by 3.
The transformation that reflects a function over the x-axis is represented by multiplying the function by -1. This gives the mirror image of the graph below the x-axis.
Let's analyze the options using these transformations:
a. f(x) = (1/3)x^2
This option involves vertical expansion by a factor of 1/3. However, we want a factor of 3, so this option is not correct.
b. f(x) = -3x^2
This option involves vertical expansion by a factor of 3 through multiplication by -3. However, it does not reflect the function over the x-axis, so this option is also not correct.
c. f(x) = -1/x^2 + 3
This option does not involve vertical expansion or reflection over the x-axis. Therefore, it is not the correct choice.
d. f(x) = -(1/3)x^2
This option involves both vertical expansion by a factor of 3 through multiplication by -1/3 and reflection over the x-axis through the negative sign. Thus, this is the correct choice.
In conclusion, the correct option is d. f(x) = -(1/3)x^2, which results in the graph of f(x) = x^2 being expanded vertically by a factor of 3 and reflected over the x-axis.