# Identify the parent function $g(x)=\sqrt {-x+2 }$

Identify the parent function

$g(x)=\sqrt {-x+2 }$

1-Identify the parent function

2-Identify the transformations being applied, in appropriate order

3-sketch a graph of the transformed function

my work for first one

from parent function

the parent function is

$f(x)=\sqrt {x }$

then

$g(x)= f(-x+2)$

Sometimes its just easier thinking about the equation inside out looking at $g^{-1}$. Consider $$\begin{array}{lll} g(x)=f(a(x-b))\\ f^{-1}(g(x))=f^{-1}(f(a(x-b)))\\ f^{-1}(g(x))=a(x-b)\\ \frac{1}{a}[f^{-1}(g(x))] + b=x\\ \end{array}$$ Which looks somewhat cryptic, but lets look at your particular example. $$\begin{array}{lll} y=\sqrt{-x+2}\\ y^2=-x+2 = -(x-2),y\ge0\\ -y^2= x-2\\ -y^2 + 2= x\\ \end{array}$$ If we graph this function with $x$ on the vertical axis and $y$ on the horizontal axis we see that we have a parabola that has been flipped about the horizontal axis and shifted upwards by 2 units. Now when we pull the switcheroo on the x/y-axes (flip the graph diagonally so the $x$ is on the horizontal axis and $y$ is on the veritcal axis) we see that the horizontal flip becomes a vertical flip and the upward shift of 2 units becomes a rightward shift of 2 units.
Try copying the image into Word, and print it out. Try holding the printout up to the light to see the "switcheroo" in action. Hold the printout up so that the black $y$ is at the top, then flip it over so that the green $y$ is at the right. Pay special attention to what happens to the position of the vertex and the direction that the parabola opens toward. Note that the dotted red curve is the negative shadow of the square root parent function; it just makes it easier to see the parabola.
• Translation $(-2,0)$ gives $\sqrt{x+2}$
• Multiplication through the y-axis by -1 gives $\sqrt{-x+2}$.
Graph: The start point is $(2,0)$. Then draw a square root graph to the left.