Summary
Highlights
The video begins by introducing the parent function, y = √x. This graph starts at the origin (0,0) and increases at a decreasing rate. The domain and range for this function are both from 0 to infinity, inclusive.
The video then explains how negative signs affect the graph. A negative sign in front of the radical (e.g., y = -√x) reflects the graph over the x-axis, causing it to point towards Quadrant 4. A negative sign inside the radical (e.g., y = √-x) reflects it over the y-axis, pointing towards Quadrant 2. Two negative signs reflect it over the origin, pointing towards Quadrant 3. A quadrant-based rule is provided: positive x and positive y lead to Quadrant 1, positive x and negative y to Quadrant 4, negative x and positive y to Quadrant 2, and negative x and negative y to Quadrant 3.
Adding or subtracting a constant outside the radical (e.g., y = √x + 2) shifts the graph vertically. Adding two shifts it up two units, while subtracting one shifts it down one unit. Adding or subtracting a constant inside the radical (e.g., y = √(x-2)) shifts the graph horizontally. Subtracting two shifts it right two units, and adding three shifts it left three units.
To create a more accurate sketch, one can plot points. For y = √x, key points are (0,0), (1,1), and (4,2). If a coefficient is placed in front of the radical (e.g., y = 2√x), the Y-values are doubled, resulting in a vertical stretch. For example, (1,1) becomes (1,2) and (4,2) becomes (4,4) relative to the origin.
The video demonstrates graphing a function with multiple transformations, such as y = √(x-1) + 2. The new 'origin' starts at (1,2) due to the shifts. From this point, one goes one unit right and one unit up, then four units right and two units up. The domain is [1, ∞) and the range is [2, ∞).
A more complex example is presented: y = 3 - √ (x-4), which can be rewritten as y = -√ (x-4) + 3. This graph is shifted four units to the right and three units up, starting at (4,3). Due to the negative sign in front of the radical, it reflects over the x-axis and points towards Quadrant 4 relative to its new starting point. From (4,3), go one unit right and one unit down to (5,2), then four units right and two units down to (8,1). The domain is [4, ∞) and the range is (-∞, 3].