To graph the function [tex]\( f(x) = 3 \sqrt{x} \)[/tex], follow these steps:
1. Understand the Function: The function [tex]\( f(x) = 3 \sqrt{x} \)[/tex] is a transformation of the basic square root function [tex]\( \sqrt{x} \)[/tex]. Here, the function is multiplied by 3, which vertically stretches the graph.
2. Identify Key Points:
- When [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 3 \sqrt{0} = 0 \)[/tex].
- When [tex]\( x = 1 \)[/tex], [tex]\( f(x) = 3 \sqrt{1} = 3 \)[/tex].
- When [tex]\( x = 4 \)[/tex], [tex]\( f(x) = 3 \sqrt{4} = 6 \)[/tex].
- When [tex]\( x = 9 \)[/tex], [tex]\( f(x) = 3 \sqrt{9} = 9 \)[/tex].
3. Determine the Domain and Range:
- Domain: Since the square root function [tex]\( \sqrt{x} \)[/tex] is only defined for [tex]\( x \geq 0 \)[/tex], the domain of [tex]\( f(x) = 3 \sqrt{x} \)[/tex] is also [tex]\( x \geq 0 \)[/tex].
- Range: The vertical stretch does not affect the range starting at 0 and going to positive infinity. So, the range of [tex]\( f(x) \)[/tex] is [tex]\( y \geq 0 \)[/tex].
4. Plot the Points: Plot the points found in step 2:
- (0, 0)
- (1, 3)
- (4, 6)
- (9, 9)
5. Draw the Graph: Connect these points with a smooth curve that represents the general shape of the square root function, vertically stretched by a factor of 3.
The graph will start from the origin (0, 0) and gradually rise, becoming less steep as [tex]\( x \)[/tex] increases, reflecting the growth of the square root function but stretched vertically by a factor of 3.
This completes the graph of the function [tex]\( f(x) = 3 \sqrt{x} \)[/tex].
