Circles, translations and deciding whether a relation is a function.
Example 1
The circle (x−4)2+(y−7)2=25 has what radius?
Show worked answer
Answer: 5
- A circle has form (x−h)2+(y−k)2=r2.
- r2=25, so r=5.
Example 2
Translate the point (5,5)4 units left and 4 units up. Enter the new coordinate as x,y.
Show worked answer
Answer: (1,9)
- Left decreases the x−coordinate: 5−4=1.
- Up increases the y−coordinate: 5+4=9.
Example 3
The relation (x−3)2+(y+3)2=9, restricted to y≥−3 is drawn below for 0≤x≤6. Does it define y as a function of x?
y
0246-6-4-20 x
upper semicircleShow worked answer
Answer: Yes: each x in the domain has exactly one y−value
- The restriction y≥−3 retains only the upper semicircle.
- Each x in [0,6] has one permitted y−value, including both endpoints.