Determine the equation of the function that has the indicated transformation applied to it.
f(x)= 3(x+1)^2-2
reflection across x-axis and stretched vertically by factor of 3
reflection: f(x)->-f(x) = -3(x+1)^2 - 2
3 stretch: f(x)->3f(x) = 3(-3(x+1)^2 - 2)
= -9(x+1)^2 - 6
oops
-(3(x+1)^2 - 2) = -3(x+1)^2 + 2
then stretch:
-9(x+1)^2 + 6
Thanks
To determine the equation of the function that has the indicated transformation applied to it, we need to understand the effects of reflection across the x-axis and vertical stretching.
1. Reflection across x-axis (negation of y-coordinate):
When a function is reflected across the x-axis, the y-coordinate of each point is negated. In this case, since the original function is f(x), the reflection across the x-axis transforms it to -f(x).
2. Vertical stretching:
To modify the function by stretching it vertically, we multiply the entire function by a factor. In this case, the original function f(x) is stretched vertically by a factor of 3, so we multiply it by 3, resulting in 3f(x).
Combining both transformations, the equation for the transformed function is:
g(x) = -3f(x)
Now substitute the original function f(x) into the equation:
g(x) = -3[3(x+1)^2-2]
Simplify the expression:
g(x) = -9(x+1)^2 + 6
Therefore, the equation of the function that has been reflected across the x-axis and stretched vertically by a factor of 3 is g(x) = -9(x+1)^2 + 6.