H(x): Horizontal shift 3 units right, vertical shift 7 units up: H(x) = f(x - 3) + 7
M(x): Horizontal shift 7 units left, vertical shift 10 units down, negate: M(x) = -f(x + 7) - 10
S(x): Reflect across the y-axis, vertical shift 8 units up: S(x) = f(-x) + 8
H(x) = (x – 3)^4 +7:
This function takes f(x) and performs two transformations:
Horizontal shift: We move the graph 3 units to the right. Imagine grabbing the graph of f(x) and sliding it 3 units to the right without changing its shape. This is achieved by replacing x with (x - 3) in f(x).
Vertical shift: We move the entire graph 7 units up. This is simply adding 7 to the transformed function, f(x - 3).
Therefore, H(x) = f(x - 3) + 7.
M(x) = - (x + 7)^4 – 10:
This function involves three transformations:
Horizontal shift: We move the graph 7 units to the left. Similar to H(x), we replace x with (x + 7) in f(x).
Negation: We flip the graph upside down. This is achieved by multiplying the transformed function, f(x + 7), by -1.
Vertical shift: We move the entire flipped graph down 10 units. Adding -10 to the negated function completes the transformation.
So, M(x) = -f(x + 7) - 10.
S(x) = (-x)^4 + 8:
This function involves two transformations:
Reflection across the y-axis: We flip the graph of f(x) horizontally. This is achieved by replacing x with (-x) in f(x).
Vertical shift: We move the entire flipped graph 8 units up. Similar to H(x) and M(x), we add 8 to the transformed function.
Therefore, S(x) = f(-x) + 8.