To determine the graph of the function [tex]\( f(x) = 3\sqrt{x} \)[/tex], we need to analyze and plot its key features. Let’s go through this step by step:
1. Function Definition:
- The function [tex]\( f(x) = 3\sqrt{x} \)[/tex] describes a transformation of the basic square root function [tex]\( \sqrt{x} \)[/tex].
- We multiply the square root by 3, which stretches the graph vertically by a factor of 3.
2. Domain:
- The square root function [tex]\( \sqrt{x} \)[/tex] is defined for all [tex]\( x \geq 0 \)[/tex].
- Therefore, the domain of [tex]\( f(x) = 3\sqrt{x} \)[/tex] is [tex]\( [0, \infty) \)[/tex].
3. Key Points:
- Let's identify a few key points to plot:
- When [tex]\( x = 0 \)[/tex], [tex]\( f(0) = 3\sqrt{0} = 0 \)[/tex].
- When [tex]\( x = 1 \)[/tex], [tex]\( f(1) = 3\sqrt{1} = 3 \)[/tex].
- When [tex]\( x = 4 \)[/tex], [tex]\( f(4) = 3\sqrt{4} = 6 \)[/tex].
- When [tex]\( x = 9 \)[/tex], [tex]\( f(9) = 3\sqrt{9} = 9 \)[/tex].
- These points will help to sketch the graph accurately.
4. Behavior:
- As [tex]\( x \)[/tex] increases, [tex]\( \sqrt{x} \)[/tex] increases but at a decreasing rate because the growth of the square root function slows down.
- Multiplying by 3 maintains this slower growth but ensures that [tex]\( f(x) \)[/tex] grows three times faster than the simple square root function.
5. Graph Sketch:
- Start by plotting the key points: (0,0), (1,3), (4,6), and (9,9).
- Connect these points smoothly, noting the characteristic shape of the square root function.
- Ensure the graph has a gentle upward curvature that flattens as [tex]\( x \)[/tex] becomes very large.
Putting these steps together, the graph of [tex]\( f(x) = 3\sqrt{x} \)[/tex]:
- Starts at the origin (0,0).
- Passes through the points (1,3), (4,6), and (9,9).
- Extends indefinitely to the right (with [tex]\( x \geq 0 \)[/tex]), gradually increasing at a decreasing rate.
- Lies only in the first quadrant since [tex]\( f(x) \)[/tex] is not defined for negative [tex]\( x \)[/tex].
This detailed analysis gives us a full understanding of how to sketch the function [tex]\( f(x) = 3\sqrt{x} \)[/tex].
