Question

1. draw a graph of the function ( f(x)=sqrt{x + 3} )

include a table of the parent function in your work.
(ms.rhee will manually grade this)

Explanation:

Step1: Identify the parent function

The parent function of \( f(x)=\sqrt{x + 3} \) is \( g(x)=\sqrt{x} \).

Step2: Create a table for the parent function \( g(x)=\sqrt{x} \)

We choose some non - negative values of \( x \) (since the square root of a negative number is not a real number for real - valued functions) and find the corresponding \( g(x) \) values.

\( x \)	\( g(x)=\sqrt{x} \)
\( 1 \)	\( \sqrt{1}=1 \)
\( 4 \)	\( \sqrt{4}=2 \)
\( 9 \)	\( \sqrt{9}=3 \)

Step3: Analyze the transformation from \( g(x) \) to \( f(x) \)

The function \( f(x)=\sqrt{x + 3} \) is a horizontal shift of the parent function \( g(x)=\sqrt{x} \). The general form of a horizontal shift for a function \( y = g(x) \) is \( y=g(x - h) \), where if \( h>0 \), the graph shifts to the right and if \( h < 0 \), the graph shifts to the left. For \( f(x)=\sqrt{x+3}=\sqrt{x-(- 3)} \), \( h=-3 \), so the graph of \( g(x)=\sqrt{x} \) shifts 3 units to the left to get the graph of \( f(x)=\sqrt{x + 3} \).

Step4: Determine the domain and key points for \( f(x) \)

For the function \( f(x)=\sqrt{x + 3} \), the expression inside the square root must be non - negative. So \( x+3\geq0\), which gives \( x\geq - 3 \).
To find key points, we can use the transformation. The point \( (0,0) \) on \( g(x) \) shifts to \( (-3,0) \) on \( f(x) \), the point \( (1,1) \) on \( g(x) \) shifts to \( (-2,1) \) on \( f(x) \), the point \( (4,2) \) on \( g(x) \) shifts to \( (1,2) \) on \( f(x) \), and the point \( (9,3) \) on \( g(x) \) shifts to \( (6,3) \) on \( f(x) \).

Step5: Sketch the graph

1. First, plot the key points of the parent function \( g(x)=\sqrt{x} \): \( (0,0) \), \( (1,1) \), \( (4,2) \), \( (9,3) \).
2. Then, apply the horizontal shift. Move each of these points 3 units to the left to get the points for \( f(x) \): \( (-3,0) \), \( (-2,1) \), \( (1,2) \), \( (6,3) \).
3. Draw a smooth curve through the shifted points, starting from \( (-3,0) \) and increasing slowly (since the square - root function has a decreasing rate of increase) as \( x \) increases.

Answer:

The parent function is \( g(x)=\sqrt{x} \) with the table as shown above. The graph of \( f(x)=\sqrt{x + 3} \) is the graph of \( g(x)=\sqrt{x} \) shifted 3 units to the left. The key points for \( f(x) \) are \( (-3,0) \), \( (-2,1) \), \( (1,2) \), \( (6,3) \) and the graph is a smooth curve passing through these points with domain \( x\geq - 3 \).