Vertex Formula: The parabola's Vertex Formula serves to determine the coordinates of the point where the parabola intersects its axis of symmetry, known as the vertex, denoted as (h, k). The standard equation for a parabola is represented as y = ax
2
+ bx + c. When the coefficient of x
2
is positive, the vertex is positioned at the lowest point of the U-shaped curve, whereas a negative coefficient places the vertex at the highest point of the U-shaped curve.
The vertex corresponds to the minimum point when the parabola opens upward or the maximum point when it opens downward, marking the turning point where the parabola changes its direction. Exploring the vertex formula further and working through examples will enhance understanding of its application in solving parabolic equations.
What is Vertex Formula
The Vertex Formula is instrumental in determining the coordinates of the vertex for a parabola.
The standard parabolic equation is expressed as y = ax
2
+ bx + c.
In the vertex form of the parabola, it's represented as y = a(x - h)
2
+ k.
The vertex coordinates (h, k) can be found using two methods:
(h,k)=(− 2a b ,− 4a D ),
where D (the discriminant) = b
2
−4ac (h,k) can be found by setting h=− 2a b and then evaluating y at h to determine the value of k.
Vertex Formula
There are two formulas used to determine the vertex:
Formula 1: (h,k)=(− 2a b ,− 4a D )
Where: D represents the denominator (h,k) are the coordinates of the vertex
Formula 2: x-coordinate of the vertex = − 2a b
Derivation of Vertex Formulas
Formula 1:
Starting with the standard form of a parabola, y = ax
2
+ bx + c, the conversion to the vertex form y = a(x - h)^2 + k is achieved by completing the square.
First, by subtracting c from both sides, we get: y - c = ax
2
+ bx
Factoring out 'a': y - c = a(x
2
+ b/a x)
Identifying half of the coefficient of x as b/2a and its square as b
2
/4a
2
,
we add and subtract this term within the parentheses on the right side:
y - c = a(x
2
+ b/a x + b
2
/4a
2
- b
2
/4a
2
)
This expression simplifies to:
y - c = a((x + b/2a)
2
- b
2
/4a
2
) E
Expanding and rearranging terms, adding 'c' to both sides:
y = a(x + b/2a)
2
– b
2
/4a + c
y = a(x + b/2a)
2
- (b
2
- 4ac) / (4a)
Comparing this with the vertex form y = a(x - h)
2
+ k, the following vertex values are derived:
h = -b/2a k = -(b
2
- 4ac) / (4a)
The term b^2 - 4ac represents the discriminant (D).
Hence, the vertex formula is:
(h,k)=(− 2a b ,− 4a D ) where D=b
2
−4ac.
Formula 2:
For those finding it challenging to memorize the earlier formula, you can simply recall the x-coordinate formula of the vertex and then substitute it into the equation y = ax
2
+ bx + c to determine the y-coordinate of the vertex. x-coordinate of the vertex (h) = − 2a b
Alternatively, if you prefer not to use either of the above formulas to locate the vertex, you can manually complete the square to transform the equation y = ax
2
+ bx + c into the form y = a(x - h)
2
+ k and manually determine the vertex (h, k).
Vertex Formula Solved Examples
Example 1:
Find the vertex of the parabola represented by the equation y=−2x 2 +8x−5.
Solution: Given equation: y=−2x 2 +8x−5
Coefficients: a=−2, b=8, c=−5
Discriminant: D=b 2 −4ac=( 8) 2 −4(−2)(−5)= 64−40=24
Using the vertex formula (Formula 1):
Vertex, (h,k)=(− 2a b ,− 4a D )
(h,k)=(− 2(−2) 8 ,− 4(−2) 24 )=(−2,3)
Therefore, the vertex of the given parabola is (−2,3).
Example 2:
Consider the parabola with the equation y=x 2 +6x+9. Find its vertex.
Solution: Given equation: y=x 2 +6x+9
Coefficients: a=1, b=6, c=9
Discriminant:
D=b 2 −4ac=(6) 2 −4(1)(9)= 36−36=0
Using the vertex formula (Formula 1):
(h,k)=(− 2a b ,− 4a D )
(h,k)=(− 2(1) 6 ,− 4(1) 0 )=(−3,0)
Hence, the vertex of the given parabola is ( − 3 , 0 ) (−3,0).
These examples demonstrate the application of the vertex formula to determine the vertex of different parabolic equations.
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