for a quadratic function, how to find the coordinates of the vertex of its graph? 2. functions and...
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For a quadratic function, how to find the coordinates of the vertex of its graph?
2.2. Functions and Graphs Functions and Graphs
(a) For y = a(x-h)2+k, the vertex of the graph is (h , k).(b) For y = ax2+bx+c, the vertex of the graph is
2ba
2 4
4b ac
a( , ). = a(x2+ ) +cb xa
= a[x2+ + - ] +cb xa
2( )2ba
2( )2ba
= a(x+ )2- +c2ba
2
4ba
= a[x- ]2+( )2ba
2 4( )
4b ac
a
k =2 4
4b ac
a
2ba
∴ h = ,
y-coordinate of Vertex Easy Memory Tips:
Since ∆=b2-4ac, we can memorize as .4a
y = ax2+bx+cTake out the coefficient of x2 in the first two terms
Add up the square of half the coefficient of xMove outside
the brackets
2( )2ba
By completing square, we can rewrite y = ax2+bx+c in the form of y = a(x-h)2+k, so as to find h and k.
kh-
For y = a(x-h)2+k, the vertex of the graph is (h , k), where
E.g.E.g.
y = x2-12x+
∴ The coordinates of the vertex of its graph are (6 , 4).
Completing Completing SquareSquare
= x2-2(6)x+62
= (x-6)2+4 k
36+440= x2-12x+36+4+4
h
k =2 4
4b ac
a
2ba
h = , .
2.2. Functions and Graphs Functions and GraphsFor a quadratic function, how to find the coordinates of the vertex of its graph?
For the function y = x2-12x+40, find the coordinates of the vertex of its graph.
E.g.E.g. For the function y = x2-12x+40, find the coordinates of the vertex of its graph.
x-coordinate = = 6122 1 =
y-coordinate = = 42( 12) 4 1 404 1
=
∴ The coordinates of the vertex of its graph are (6 , 4).
Algebraic Algebraic MethodMethod
144 1604 1
The answer is the same as using completing square, but using the formula is more efficient.
2.2. Functions and Graphs Functions and Graphs
For y = a(x-h)2+k, the vertex of the graph is (h , k), where
k =2 4
4b ac
a
2ba
h = , .2ba
2 4
4b ac
a
For a quadratic function, how to find the coordinates of the vertex of its graph?
E.g.E.g. For y = 2x2+2x-1, find the coordinates of the vertex of its graph.
y = 2x2+2x-1
= 2(x2+x)-1
= 2[x2+x+21( )
2
= 2(x+ )2-12
32
∴ The coordinates of the vertex of its graph
Take out the coefficient of x2 in the first two terms
Add up the square of half the coefficient of x
21( )2
Take out from the bracket
21( )2
2 2
12
32
]
= 2[x - ]2-1( )2
32 k
h 12
32
are ( , ).
2 + [ ]-1
2.2. Functions and Graphs Functions and Graphs
Completing Completing SquareSquare
For y = a(x-h)2+k, the vertex of the graph is (h , k), where
k =2 4
4b ac
a
2ba
h = , .
For a quadratic function, how to find the coordinates of the vertex of its graph?
E.g.E.g. For y = 2x2+2x-1, find the coordinates of the vertex of its graph.
x-coordinate = = 12
22 2
=
y-coordinate = = 32
22 4 2 ( 1)
4 2 =
∴ The coordinates of the vertex of the graph are
4 84 2
12
32
( , ).
2.2. Functions and Graphs Functions and Graphs
Algebraic Algebraic MethodMethod
The answer is the same as using completing square, but using the formula is more efficient.
For y = a(x-h)2+k, the vertex of the graph is (h , k), where
k =2 4
4b ac
a
2ba
h = , .2ba
2 4
4b ac
a
For a quadratic function, how to find the coordinates of the vertex of its graph?
E.g.E.g.
x-coordinate = = 12
22 2
=
∴ The coordinates of the vertex of the graph are ( , ).12
32
2ba
It is hard to memorize the formula for the y-coordinate. We can find the x-coordinate first and then substitute it into the equation to find the y-coordinate.
24 -1-13
22x2-2x+1
Substitute x= into the equation, we have12
12
2( )2+2( )-112
y =
2.2. Functions and Graphs Functions and Graphs
Algebraic Algebraic MethodMethod
For y = 2x2+2x-1, find the coordinates of the vertex of its graph.
For y = a(x-h)2+k, the vertex of the graph is (h , k), where
k =2 4
4b ac
a
2ba
h = , .
For a quadratic function, how to find the coordinates of the vertex of its graph?