The graph of g(x) is the graph of f(x) reflected over the x-axis, translated 6 units to the left and shifted down 3 units. Write the equation for g(x)
Well, if I were to write the equation for g(x) in a clownishly funny way, it would go something like this:
g(x) = -(f(x+6)) - 3
Basically, g(x) is f(x) but turned upside down like a disoriented clown, then moved awkwardly 6 units to the left, as if trying to avoid stepping on a banana peel, and finally shifting down 3 units like it just heard a bad joke.
To obtain the equation for g(x) from the equation of f(x), we can apply the given transformations step by step.
1. Reflecting over the x-axis: To reflect a function over the x-axis, we change the sign of the entire equation. Therefore, the equation becomes -f(x).
2. Translating 6 units to the left: To shift a function k units to the left, we replace x with (x + k) in the equation. So, the equation becomes -f(x + 6).
3. Shifting down 3 units: To shift a function h units downward, we subtract h from the equation. Thus, the equation becomes -f(x + 6) - 3.
Therefore, the equation for g(x) is -f(x + 6) - 3.
To create a new function g(x) from a given function f(x) with a reflection over the x-axis, translation to the left, and downward shift, you need to apply three transformations: a reflection, a translation, and a shift.
1. Reflection over the x-axis: To reflect a function over the x-axis, you need to negate the entire function. For example, if f(x) = 2x + 5, then the reflection would be -f(x) = -(2x + 5) = -2x - 5.
2. Translation to the left: To translate a function to the left, you need to subtract a constant value from the x-coordinates. In this case, the translation is 6 units to the left, so you would replace x with (x + 6) in the function obtained from the reflection.
Using the function obtained in step 1: -2x - 5, the translation would be: -2(x + 6) - 5.
3. Downward shift: To shift the function downward, you need to subtract a constant value from the y-coordinates. In this case, the shift is 3 units down, so you subtract 3 from the function obtained from the translation.
Using the function obtained in step 2: -2(x + 6) - 5, the downward shift would be: -2(x + 6) - 5 - 3.
Combining all these transformations, the equation for g(x) is:
g(x) = -2(x + 6) - 5 - 3
= -2x - 12 - 5 - 3
= -2x - 20
Therefore, the equation for g(x) is g(x) = -2x - 20.
reflection: -f(x)
6 left: -f(x+6)
3 down: -f(x+6) - 3