To determine which graph shows the function [tex]\( C(x) = \frac{1}{4} \sqrt{x} + 2 \)[/tex], we can start by calculating a few values of [tex]\( C(x) \)[/tex] for corresponding values of [tex]\( x \)[/tex]. This helps us to understand the behavior of the function and to sketch the graph accurately.
1. Value at [tex]\( x = 0 \)[/tex]:
[tex]\[
C(0) = \frac{1}{4} \sqrt{0} + 2 = 0 + 2 = 2
\][/tex]
So, [tex]\( C(0) = 2 \)[/tex].
2. Value at [tex]\( x = 0.2 \)[/tex]:
[tex]\[
C(0.2) = \frac{1}{4} \sqrt{0.2} + 2 \approx \frac{1}{4} \times 0.447 + 2 \approx 0.11175 + 2 = 2.11175
\][/tex]
Roughly, [tex]\( C(0.2) \approx 2.11175 \)[/tex].
3. Value at [tex]\( x = 0.4 \)[/tex]:
[tex]\[
C(0.4) = \frac{1}{4} \sqrt{0.4} + 2 \approx \frac{1}{4} \times 0.632 = 0.158 + 2 = 2.158
\][/tex]
4. Value at [tex]\( x = 0.6 \)[/tex]:
[tex]\[
C(0.6) = \frac{1}{4} \sqrt{0.6} + 2 \approx \frac{1}{4} \times 0.775 + 2 = 0.19375 + 2 = 2.19375
\][/tex]
5. Value at [tex]\( x = 0.8 \)[/tex]:
[tex]\[
C(0.8) = \frac{1}{4} \sqrt{0.8} + 2 \approx \frac{1}{4} \times 0.894 + 2 = 0.2235 + 2 = 2.2235
\][/tex]
6. Value at [tex]\( x = 1.0 \)[/tex]:
[tex]\[
C(1.0) = \frac{1}{4} \sqrt{1.0} + 2 = \frac{1}{4} \times 1.0 + 2 = 0.25 + 2 = 2.25
\][/tex]
So, now we have the following points:
[tex]\[
(0, 2), (0.2, 2.11175), (0.4, 2.158), (0.6, 2.19375), (0.8, 2.2235), (1.0, 2.25)
\][/tex]
7. Value at [tex]\( x = 1.2 \)[/tex]:
[tex]\[
C(1.2) = \frac{1}{4} \sqrt{1.2} + 2 \approx \frac{1}{4} \times 1.10 + 2 = 0.27435 + 2 = 2.27435
\][/tex]
8. Value at [tex]\( x = 1.4 \)[/tex]:
[tex]\[
C(1.4) = \frac{1}{4} \sqrt{1.4} + 2 \approx \frac{1}{4} \times 1.183 + 2 = 0.296 + 2 = 2.296
\][/tex]
9. Value at [tex]\( x = 1.6 \)[/tex]:
[tex]\[
C(1.6) = \frac{1}{4} \sqrt{1.6} + 2 \approx \frac{1}{4} \times 1.265 + 2 = 0.31625 + 2 = 2.31625
\][/tex]
10. Value at [tex]\( x = 1.8 \)[/tex]:
[tex]\[
C(1.8) = \frac{1}{4} \sqrt{1.8} + 2 \approx \frac{1}{4} \times 1.34 + 2 = 0.335 + 2 = 2.335
\][/tex]
Plotting some of these points:
- (0, 2)
- (0.2, 2.11175)
- (0.4, 2.158)
- (0.6, 2.19375)
- (0.8, 2.2235)
- (1.0, 2.25)
- (1.2, 2.27435)
- (1.4, 2.296)
- (1.6, 2.31625)
- (1.8, 2.335)
We observe that the function starts at [tex]\( y = 2 \)[/tex] and increases slowly as [tex]\( x \)[/tex] increases. The curve is relatively shallow and grows slowly due to the square root component.
Thus, the graph that represents this data should start at [tex]\( y = 2 \)[/tex] when [tex]\( x = 0 \)[/tex] and gradually increase as [tex]\( x \)[/tex] increases. The curve's shape should align with the points we've calculated. From the description and behavior, the possible options should align closely with these trends and points.
