Combined Transformations of Square Root Functions
Combined transformations of square root functions is a Grade 11 Algebra 1 topic from enVision Chapter 10 covering the general form g(x) = sqrt(x - h) + k. The parameter h produces a horizontal shift (right if positive) and k produces a vertical shift (up if positive). The domain becomes x >= h and range becomes y >= k, with starting point at (h, k). For example, g(x) = sqrt(x - 3) + 2 has domain x >= 3, range y >= 2, and starts at (3, 2). The function h(x) = sqrt(x + 1) - 4 starts at (-1, -4). Both shifts apply simultaneously.
Key Concepts
The general form of a transformed square root function is $g(x) = \sqrt{x h} + k$, where $h$ represents horizontal translation and $k$ represents vertical translation. The domain is $x \geq h$ and the range is $y \geq k$, with the starting point at $(h, k)$.
Common Questions
What is the general form of a transformed square root function?
g(x) = sqrt(x - h) + k, where h is the horizontal shift and k is the vertical shift. The starting point is (h, k).
What are the domain and range of g(x) = sqrt(x - 3) + 2?
Domain: x >= 3. Range: y >= 2. The starting point is (3, 2), shifted right 3 and up 2 from the origin.
What is the starting point of h(x) = sqrt(x + 1) - 4?
(-1, -4). Rewrite as sqrt(x - (-1)) + (-4). h = -1 shifts left 1, k = -4 shifts down 4.
How do you identify the domain from the transformed form?
Set x - h >= 0 and solve: x >= h. The domain always starts at x = h.
How does f(x) = sqrt(x - 5) + 1 compare to the parent function?
It shifts the parent function right 5 units and up 1 unit. The graph has the same shape but starts at (5, 1) instead of (0, 0).
Can h be negative in g(x) = sqrt(x - h) + k?
Yes. When h is negative, subtracting a negative number means adding, so x - (-2) = x + 2 shifts the graph left 2 units.