Graphing a Square Root Function

The table below indicates the shape of the curve according to the role of parameters |\color{#3B87CD}{a}| and |\color{#EC0000}{b}| of the square root function.

	|\color{#EC0000}{b}<0|	|\color{#EC0000}{b}>0|
|\color{#3B87CD}{a}>0|	The function is defined above and to the left of the vertex.	The function is defined above and to the right of the vertex.
|\color{#3B87CD}{a}<0|	The function is defined below and to the left of the vertex.	The function is defined below and to the right of the vertex.

1. Place the vertex at the coordinates |(\color{#51B6C2}{0},\color{#FA7921}{0})| and determine the shape of the curve by analyzing parameters |\color{#3B87CD}{a}| and |\color{#EC0000}{b}.|

2. Calculate and plot additional points.

3. Graph the square root function.

Graph the function |f(x)=2\sqrt{-3x}.|

2. Calculate and place additional points
   
   First choose random |\color{#3a9a38}{x}| values and calculate the corresponding |\color{#3a9a38}{f(x)}| values. Since the curve is to the left of the vertex, choose negative |\color{#3a9a38}{x}| values. First, taking |\color{#3a9a38}{x}=\color{#3a9a38}{-2}.| ||\begin{align}f(\color{#3a9a38}{x})&=2\sqrt{-3\color{#3a9a38}{x}}\\f(\color{#3a9a38}{-2})&=2\sqrt{-3(\color{#3a9a38}{-2}) }\\f(\color{#3a9a38}{-2})&=2\sqrt{6}\\\color{#3a9a38}{f(-2)}&\approx\color{#3a9a38}{4 .9}\\\end{align}||
   The more points with known coordinates, the more precise the sketch of the curve. Therefore, |\color{#3a9a38}{x}=\color{#3a9a38}{-3}| and |\color{#3a9a38}{x}=\color{#3a9a38}{-12}| give the points |(-3,6)| and |(-12,12).|

When choosing |x|-values to place additional points on the Cartesian plane, ensure the result under the root is a square number. It avoids approximating coordinates.

In the previous example, |x=-2| is not the best choice, because |-3\times-2=6.| Indeed, |6| is not a square number, because |\sqrt{6}\approx245.| However, |x=-3| and |x=-12| result in |9| and |36| respectively, two square numbers under the root.

1. Place the vertex at the coordinates |(\color{#51B6C2}{h},\color{#FA7921}{k})| and analyze parameters |\color{#3B87CD}{a}| and |\color{#EC0000}{b}| to determine the shape of the curve.

2. Calculate and plot additional points.

3. Graph the square root function.

In step 3, although it is not always possible, calculate the value of the zero (the |x|-intercept) and the |y|-intercept.

Graph the function |f(x)=-3\sqrt{x-2}+6.|

2. Calculate and place additional points
   
   It can be determine from the previous image that all the potential curves cross the |x|-axis. It implies the function has an |x|-intercept. However, there is no |y|-intercept because none of the curves cross the |y|-axis. Now, replace |f(x)| with |0| to calculate the |x|-intercept. ||\begin{align}\color{#3a9a38}{f(x)}&=-3\sqrt{x-2}+6\\\color{#3a9a38 }{0}&=-3\sqrt{x-2}+6\\-6&=-3\sqrt{x-2}\\2&=\sqrt{x-2}\\4&=x-2\\6&=x\\\end{align}||
   Therefore, the curve has a zero at |(6,0).| The function |f(x)=-3\sqrt{x-2}+6| does not have a |y|-intercept. Replacing |x| with |0| gives |\sqrt{-2},| which is not possible.
   
   Now, calculate the coordinates of a few additional points to sketch a more precise function. Since the curve is to the right of the vertex, choose |x|-values greater than |2.| First, we can use |x=11.|
   ||\begin{align}f(\color{#3a9a38}{x})&=-3\sqrt{\color{#3a9a38}{x}-2}+ 6\\f(\color{#3a9a38}{11})&=-3\sqrt{\color{#3a9a38}{11}-2}+6\\f(11)&=-3\sqrt{9 }+6\\f(11)&=-3\times3+6\\\color{#3a9a38}{f(11)}&=\color{#3a9a38}{-3}\\\end{align}|| Similarly, with |x=18,| it gives another point with the coordinates |(18,-6).|