Circles And Circular Functions Worksheet - Dr. Neal, Math 116 - 2008 Page 2
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5Dr. Neal, Fall 2008
Upper and Lower Semicircle Functions
2
2
2
+ y
= r
The circle x
defines two semicircle functions, which are the top-half and
lower-half of the circle. These functions are obtained y solving for y :
2
2
2
2
2
2
2
2
+ y
= r
→ y
= r
− x
→ y = ± r
− x
x
–r
r
r
-r
–r
r
2
2
y = r
− x
2
2
y = − r
− x
Domain: − r ≤ x ≤ r
Range: 0 ≤ y ≤ r
Domain: − r ≤ x ≤ r
Range: − r ≤ y ≤ 0
Upper Semi-circular Function
Lower Semi-circular Function
Example 2. (a) Give the equation of the upper-semicircle function centered at the origin
with radius 6.
(b) Graph the function and state its domain and range.
(c) Solve for the x that make y = 4 .
(d) For which x is y ≥ 4 and for which x is y < 4?
Solution. (a) Using r = 6, the entire circle
y
2
2
+ y
= 36. So the upper-
has equation x
6
2
semi-circle function is y = 36 − x
.
4
2
2
2
(c) If y = 4 , then x
+ 4
= 36 → x
= 20
→ x = ± 20 = ± 4 × 5 = ± 2 5
− 20
–6
6
20
Domain: −6 ≤ x ≤ 6
Range: 0 ≤ y ≤ 6
(d) From the graph, we see that y ≥ 4 when − 20 ≤ x ≤ 20 . We see that y < 4 when
−6 ≤ x < − 20 or when
20 < x ≤ 6 .
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