To determine the graph of the function [tex]\( f(x) = \sqrt{\frac{1}{4} x} \)[/tex], let's go through the process step-by-step.
1. Understanding the Function:
- The function [tex]\( f(x) = \sqrt{\frac{1}{4} x} \)[/tex] can be rewritten as [tex]\( f(x) = \sqrt{\frac{x}{4}} \)[/tex] or [tex]\( f(x) = \frac{\sqrt{x}}{2} \)[/tex].
- This function involves a square root, indicating that it will only provide real outputs for non-negative inputs ([tex]\(x \geq 0\)[/tex]).
2. Generating Values:
- To get a sense of the function's shape, we can compute some values of [tex]\( f(x) \)[/tex] for different [tex]\( x \)[/tex] values.
3. Values of [tex]\( x \)[/tex] and Corresponding [tex]\( f(x) \)[/tex]:
- Let's consider [tex]\( x \)[/tex] values ranging from 0 to 10. This range is chosen to see how the function behaves over a reasonable interval.
- We'll take 100 equally spaced points between 0 and 10.
Below is a table of some sampled values from the function for illustration purposes:
| [tex]\( x \)[/tex] | [tex]\( f(x) \)[/tex] |
|----------------|---------------------------|
| 0.0 | 0.0 |
| 0.1010101 | 0.15891043 |
| 0.2020202 | 0.22473329 |
| 0.3030303 | 0.27524094 |
| 0.4040404 | 0.31782086 |
| 0.50505051 | 0.35533453 |
| 0.60606061 | 0.38924947 |
| 0.70707071 | 0.42043748 |
| 0.80808081 | 0.44946657 |
| 0.90909091 | 0.47673129 |
| 1.01010101 | 0.50251891 |
| 1.11111111 | 0.52704628 |
| 1.21212121 | 0.55048188 |
| 1.31313131 | 0.57295971 |
| 1.41414141 | 0.59458839 |
| ... | ... |
| 9.6969697 | 1.55699789 |
| 9.7979798 | 1.56508624 |
| 9.8989899 | 1.57313301 |
| 10.0 | 1.58113883 |
4. Graph Interpretation:
- The graph of [tex]\( f(x) = \sqrt{\frac{1}{4} x} \)[/tex] will be a curve that starts at the origin (0,0) and rises gradually as [tex]\( x \)[/tex] increases.
- The function [tex]\( f(x) \)[/tex] increases at a decreasing rate because the square root function grows slower as [tex]\( x \)[/tex] gets larger.
By plotting these points, you will obtain a smooth curve that starts at the origin and curves upwards as [tex]\( x \)[/tex] increases. This visual representation matches the characteristics of the square root function, scaled appropriately by the factor of [tex]\( \frac{1}{2} \)[/tex].
