Question

use transformations of ( f(x) = sqrt{x} ) to graph the following function. ( g(x) = 2sqrt{x - 5} + 4 )

Explanation:

Step1: Analyze horizontal shift

For the function \( f(x)=\sqrt{x} \), the transformation \( \sqrt{x - 5} \) represents a horizontal shift. The rule for horizontal shift is: if we have \( f(x - h) \), the graph shifts \( h \) units to the right when \( h>0 \). Here, \( h = 5 \), so the graph of \( \sqrt{x} \) shifts 5 units to the right to get \( \sqrt{x - 5} \).

Step2: Analyze vertical stretch

The coefficient 2 in front of \( \sqrt{x - 5} \) (i.e., \( 2\sqrt{x - 5} \)) represents a vertical stretch. The rule for vertical stretch is: if we have \( a\cdot f(x) \) with \( a>1 \), the graph is stretched vertically by a factor of \( a \). Here, \( a = 2 \), so the graph of \( \sqrt{x - 5} \) is stretched vertically by a factor of 2.

Step3: Analyze vertical shift

The \( + 4 \) at the end (i.e., \( 2\sqrt{x - 5}+4 \)) represents a vertical shift. The rule for vertical shift is: if we have \( f(x)+k \), the graph shifts \( k \) units up when \( k>0 \). Here, \( k = 4 \), so the graph of \( 2\sqrt{x - 5} \) shifts 4 units up.

To graph \( g(x)=2\sqrt{x - 5}+4 \) from \( f(x)=\sqrt{x} \):

1. Start with the parent function \( y = \sqrt{x} \), which has a domain \( x\geq0 \) and starts at the origin \( (0,0) \), passing through \( (1,1) \), \( (4,2) \), etc.
2. Shift the graph of \( y=\sqrt{x} \) 5 units to the right: the new key points will be \( (5,0) \), \( (6,1) \), \( (9,2) \), etc. (since replacing \( x \) with \( x - 5 \) shifts right 5).
3. Stretch the graph vertically by a factor of 2: multiply the \( y \)-coordinates of the shifted points by 2. So the points become \( (5,0\times2)=(5,0) \), \( (6,1\times2)=(6,2) \), \( (9,2\times2)=(9,4) \), etc.
4. Shift the graph 4 units up: add 4 to the \( y \)-coordinates of the stretched points. So the final key points are \( (5,0 + 4)=(5,4) \), \( (6,2+4)=(6,6) \), \( (9,4 + 4)=(9,8) \), etc. Plot these points and draw the curve.

Answer:

To graph \( g(x)=2\sqrt{x - 5}+4 \) from \( f(x)=\sqrt{x} \):

1. Shift 5 units right: Replace \( x \) with \( x - 5 \) (transforms \( \sqrt{x} \) to \( \sqrt{x - 5} \), shifting the graph right 5 units).
2. Vertical stretch by factor 2: Multiply the function by 2 (transforms \( \sqrt{x - 5} \) to \( 2\sqrt{x - 5} \), stretching the graph vertically by 2).
3. Shift 4 units up: Add 4 to the function (transforms \( 2\sqrt{x - 5} \) to \( 2\sqrt{x - 5}+4 \), shifting the graph up 4 units).

Key points transformation:

• Parent \( f(x)=\sqrt{x} \): \( (0,0) \), \( (1,1) \), \( (4,2) \).
• After right shift (\( \sqrt{x - 5} \)): \( (5,0) \), \( (6,1) \), \( (9,2) \).
• After vertical stretch (\( 2\sqrt{x - 5} \)): \( (5,0) \), \( (6,2) \), \( (9,4) \).
• After vertical shift (\( 2\sqrt{x - 5}+4 \)): \( (5,4) \), \( (6,6) \), \( (9,8) \).

Plot these final points and draw the curve, ensuring the domain is \( x\geq5 \) (since \( x - 5\geq0\Rightarrow x\geq5 \)).