Answer:
[tex]y = -3\sqrt{x - 1} + 2[/tex]
Step-by-step explanation:
The parent square root function y = √x begins at the origin (0, 0) and is contained in quadrant I of the Cartesian coordinate system. Since x has to be non-negative for the function to be defined, as x approaches infinity, the function value f(x) also approaches infinity.
The general form of the transformation equation for a square root function is:
[tex]y = a\sqrt{x - h} + k[/tex]
where:
- a is the vertical stretch/compression factor. If a < 0, the basic curve reflects across the x-axis.
- h is the horizontal translation (right is positive, left is negative).
- k is the vertical translation (up is positive, down is negative).
The graphed function begins at (1, 2). Therefore, h = 1 and k = 2:
[tex]y = a\sqrt{x - 1} + 2[/tex]
To find the value of a, substitute the given point on the curve (2, -1) into the equation:
[tex]-1 = a\sqrt{2 - 1} + 2\\\\\\-1=a\sqrt{1}+2\\\\\\-1=a+2\\\\\\a=-3[/tex]
Therefore, the equation of the graphed function is:
[tex]y = -3\sqrt{x - 1} + 2[/tex]
