To determine the new vertex of the function \( f(x) = x^2 \) after the specified transformations, we start with the vertex of the original function, which is at the point \( (0, 0) \).
#1 Positive horizontal shift 4 units:
A horizontal shift of 4 units to the right changes the function to: \[ f(x) = (x - 4)^2 \] The vertex after this shift will be: \[ (4, 0) \]
#2 Positive vertical shift 5 units:
A vertical shift of 5 units upward changes the function to: \[ f(x) = x^2 + 5 \] The vertex after this shift will be: \[ (0, 5) \]
#3 Negative horizontal shift 3 units:
A horizontal shift of 3 units to the left changes the function to: \[ f(x) = (x + 3)^2 \] The vertex after this shift will be: \[ (-3, 0) \]
#4 Negative vertical shift 2 units:
A vertical shift of 2 units downward changes the function to: \[ f(x) = x^2 - 2 \] The vertex after this shift will be: \[ (0, -2) \]
Summary of New Vertices:
- After positive horizontal shift 4 units: (4, 0)
- After positive vertical shift 5 units: (0, 5)
- After negative horizontal shift 3 units: (-3, 0)
- After negative vertical shift 2 units: (0, -2)