(-2, 1) r 2 = 16 r = 4 44 4 4 (0, 0) r 2 = 25 r = 5 55 5 5 4 3 4 3 4 3 5 special pythagorean triple
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Center @ (-2, 1)r2 = 16r = 4
44
4
4
Center @ (0, 0)r2 = 25r = 5
55
5
5
222
222
543
cba
43
434
3
5
Special Pythagorean Triple
(___, 0)Substitute in 0 for y and solve for x.
16102 22 x 1612 2 x
152 2 x
152 x
152 x
0,152
0,152;0,152
(0, ___)Substitute in 0 for x and solve for y.
16120 22 y 1614 2 y
121 2 y121 y
321y
321,0
321,0;321,0
( h, k ) ( x, y )
222 rkyhx Substitute in h and k as 1 and -2.
222
222
21
21
ryx
ryx
Substitute in x and y as 4 and 2.
222 2214 r Solve for r2.
2
222
25169
43
r
r
222 21 ryx
2521 22 yx
022 cbyaxyxGENERAL FORM OF THE EQUATION OF A CIRCLE:
096422 yxyxGraph 222 rkyhx Convert to by completing the square. Group x terms and y terms together
and move the constant to the other side.964 22 yyxx
Complete the square of the x’s and y’s. ______9___6___4 22 yyxx
yx yx
432 22 yx
94 (+2)2 (-3)2
Center @ (-2, 3) r2 = 4r = 2
Graph 02481222 22 yxyxDivide everything by 2. Why?
0124622 yxyx ______12___4___6 22 yyxx
yx yx 2523 22 yx
49 (-3)2 (-2)2
Center @ (3, 2) r2 = 25r = 5
Focus
Directrix
-a
a
-a
yx xa y (-a)0 x
2222 0 ayayx Square both sides to remove radical.
22
222 ayayx FOIL the binomials.
22222 22 aayyaayyx
Cancel like terms on each side.ayayx 222
Solve for x2.
ayx 42
a
a2a2a2a
4a4a
Graph the following equations.
xy 122 The y is squared and the coefficient on the x is positive, the parabola opens to the right. 4a = 12, a = 3 and the vertex is at (0, 0).
6
F
6
V 3
x = - 3
Graph the following equations.
yx 162 The x is squared and the coefficient on the y is negative, the parabola opens down. 4a = -16, a = -4 and the vertex is at (0, 0).
8F 8
4
V
y = 4
Graph the following equations.
xy 82 The y is squared and the coefficient on the x is negative, the parabola opens to the left. 4a = -8, a = -2 and the vertex is at (0, 0).
4
F 4
V 2
x = 2
Graph the following equations.
182 2 yx The x is squared and the coefficient on the y is positive, the parabola opens up. 4a = 8, a = 2 and the vertex is at (2, -1).
4 F 4
V
2
y = - 3
Graph the following equations.
017422 xyy We need to complete the square of the y-terms to put in graphing form. Isolate the y-terms.
___174___22 xyy 1(-1)2
1641 2 xy Factor out the 4 as the GCF.
441 2 xy The y is squared and the coefficient on the x is positive, the parabola opens to the right. 4a = 4, a = 1 and the vertex is at (4, 1).
2
2V F
x = 3
Graph the following equations.01462 yxx
2
V
2F
We need to complete the square of the x-terms to put in graphing form. Isolate the x-terms.
___14___62 yxx 9(+3)2
843 2 yx Factor out the 4 as the GCF.
243 2 yx The x is squared and the coefficient on the y is positive, the parabola opens up. 4a = 4, a = 1 and the vertex is at (-3, -2).
y = - 3
Draw a rough graph.
(2,3)
Equation format is ...axy 42
…plug in x & y to solve for 4a. 243 2 a
a
a
429
249
xy292
Draw a rough graph.
V F
Equation format is ... hxaky 42
…distance from V to F is 1, a = 1, and plug in the vertex values.
1142 2 xy 142 2 xy
Draw a rough graph.F(-4, 4)
Equation format is ...
kyahx 42
1344 2 yx 1124 2 yx
V(-4, ?)
y = -2
…distance from F to the directrix line is 6, V is halfway, so a = 3. Plug in a and the vertex values.
4 – 3 = 1V(-4, 1)
3
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