Find the domain and range of the function y = 4(x+3/2)² - 49?

The domain of parabola is R.

y = a(x - h) + k, vertex (h, k)

The range starts from y value of vertex to +∞, for U shape parabolas, i.e. for a > 0

and [-∞, y] for upsides down parabolas.

Vertex: (-3/2, -49)

The parabola opens upward because of +4

=> Range: [-49, +∞)

Any quadratic function, its domain is going to be negative infinity to infinity. For range, if x^2 is positive (as it is in this case) it's going to be the y value of the vertex to infinity. If x^2 is negative, then it's the negative infinity to the y value of the vertex. In this case x is positive so the range is -49 to infinity.

You have the vertex form of a quadratic function, and all quadratic functions have domains of all real numbers x unless specifically restricted.

The range is determined by two things: the multiplier of the squared factor and the constant term. The constant is the maximum or minimum value of y for the function as a whole, and the multiplier of the squared factor is positive -- that means you're approaching infinity on both ends. Therefore the range is all real numbers y at least -49 [-49, oo).