Circle
X intercept
The equation of a circle with centre the origin and radius r can be
found using Pythagoras’ theorem:
The point 𝑃(𝑥, 𝑦) lies on the circle. The distance OP is the radius.
Y intercept
y
𝑂𝑃 = 𝑟
√(𝑥 − 0)2 + (𝑦 − 0)2 = 𝑟
Domain:
(𝑥 − 0)2 + (𝑦 − 0)2 = 𝑟 2
x
Range:
𝑥2 + 𝑦2 = 𝑟2
NOTE:
𝑥2 + 𝑦2 = 𝑟2
y
x
y
(𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟 2
Domain:
x
Range:
Examples: a) 𝑥 2 + 𝑦 2 = 144
Domain:
Range:
b)
(𝑥 + 2)2 + (𝑦 − 3)2 = 16
Domain:
Range:
X intercept
Semi-circle
y
𝑦 = √𝑟 2 − 𝑥 2
Y intercept
𝑦 = −√𝑟 2 − 𝑥 2
y
x
x
Domain:
Examples: a) 𝑦 = √4 − 𝑥 2
Range:
b) 𝑦 = −√9 − 𝑥 2
NOTE:
Further circles
Example 1:
𝑥 2 + 𝑦 2 − 6𝑥 + 8𝑦 = 0
Exercise 3G p 95 Q1, 2, 10, 11, 12
Square root graphs
The function 𝑦 = √𝑥 is the upper half of a parabola
Domain:
Range:
y
x
Squaring both sides gives
But the √ means take the positive square root, otherwise we don’t
have a function
𝑦 = −√𝑥
y
Domain:
Range:
x
Exercise 3G Q8, 14, 17