Chapter 7.5, Problem 73STP

To determine

To Graph:

  $y\le \sqrt{x-2}$

Explanation of Solution

Concept Used :

SQUARE ROOT FUNCTION:

  $\begin{array}{l}\text{Parent Function: }f(x)=\sqrt{x}\\ \text{Domain: {x|x}\ge \text{0}}\\ \text{Range: {f(x)|f(x)}\ge \text{0}}\\ \text{Intercepts: x=0,f(x)=0}\\ \text{Not Defined: x<0}\end{array}$

Graph:

  

TRANSFORMATIONS OF SQUARE ROOT FUNCTION:

  $\begin{array}{l}f(x)=a\sqrt{x-h}+k\\ h-\text{ Horizontal Translation}\\ ·\left|h\right|\text{ units right if h is positive }\\ ·\left|h\right|\text{ units left if h is negative}\\ \text{The Domain is {x|x}\ge \text{h}}\\ \\ k-\text{ Vertical Translation }\\ ·\left|k\right|\text{ units up if k is positive}\\ ·\left|k\right|\text{ units down if k is negative}\\ \text{The Range is {f(x)|f(x)}\ge k\}\\ \\ a-\text{ Orientation and Shape }\\ ·\text{If a<0, the graph is reflected across x-axis}\\ ·\text{If }\left|a\right|>1,\text{ the graph is vertically expanded}\\ ·\text{If 0<}\left|a\right|<1,\text{ the graph is vertically compressed}\end{array}$

Calculation:

Given: $y\le \sqrt{x-2}$

$\begin{array}{l}\text{First , we graph the boundary :}\\ \text{y = }\sqrt{x-2}\\ \text{Domain = {x|x-2}\ge \text{0}={x|x}\ge \text{2}}\\ \text{Comparing it with y=a}\sqrt{x-h}+k\\ h=2,k=0,a=1\\ \text{The minimum point is at (h,k) = (2,0)}\text{.}\\ \text{Make a table of values for x}\ge \text{2 , and graph }\\ \text{the function }\text{.}\end{array}$

x	y
2	0
3	1
6	2
11	3

$\begin{array}{l}\text{Because y is less than , the shaded region should be }\\ \text{below the boundary and within the domain , for the }\\ \text{graph of y}\le \sqrt{x-2}\end{array}$

To check , let us take any point from the shaded region and check whether it satisfies the equation of the graph or not:

  $\begin{array}{l}\text{Let (x,y)=(3,-1)}\\ \text{We have , }\\ \text{Left hand side of the equation : y = -1}\\ \text{Right hand side of the equation: }\sqrt{x-2}=1\\ \text{And , -1 < 1}\\ \text{So, the point in the shaded region satisfies}\\ \text{the equation : y }\le \sqrt{x-2}\end{array}$

Hence, the graph is as shown.