Converting a quadratic function from polynomial form to vertex form.
Goal: to convert a polynomial like y = ax2+bx+c to the form y-k = a(x-h)2 which has vertex (h,k).
Example 1: Convert y= 5x2-6x-3 to vertex form.
Step
Why and What
2
1 y= 5x -6x-3
Given
2 y = 5x2-6x
-3
Move the constant term far to the RHS
6
3 y = 5[ x2- x
Factor the coefficient of x2 from the 1st 2 terms of the
]
-3
5
RHS. Use “BIG PARENTHESES, or ‘[ ]’ .
6
3 2
3 2
2
4 y = 5[ x - x + ( ) – ( ) ] - 3 Complete the square INSIDE the “[ ]”.
5
5
5
6
6
3
(a) Half of 5 = 10 = 5
(b) Square the result. Add and subtract it inside the
“[ ]”. You have added ZERO to the RHS.
3 2
3 2
5 y = 5[( x – )
– (5) ] - 3 Replace the first 3 terms inside the “[ ]” by the square
5
of a binomial.
3 2
3 2
6 y = 5(x – )
– 5 (5)
- 3 Distribute the coefficient (5) to the two terms inside
5
the “[ ]”.
3 2 9 15
3 2 24
7 y = 5(x – ) - - = 5(x– ) Simplify the term on the right.
24
8
y+
9
y- -
5
= 5(x -
5
24
5
= 5(x –
10 Vertex = (
3
5
,-
5 5
3 2
)
5
3 2
5
24
5
)
5
5
Move the constant term on the right to the LHS.
= ANSWER
). Steepness = 5.
Rewrite in vertex form (need “-“ signs).
Find the vertex and the steepness
Example 2: Convert y= ax2 + bx + c to vertex form.
Step
Why and What
2
1 y= ax +bx+c
Given
2
2 y = ax +bx
+c
Move the constant term far to the RHS
3 y = a[ x2- 𝑏x
Factor the coefficient of x2 from the 1st 2 terms of
]
+
c
𝑎
the RHS. Use “BIG PARENTHESES, or ‘[ ]’ .
𝑏
𝑏
𝑏
4 y = a[ x2 - x + ( )2 – ( )2 ] + c
Complete the square INSIDE the “[ ]”.
𝑎
2𝑎
2𝑎
𝑏
𝑏
(a) Half of 𝑎 = 2𝑎
(b) Square the result. Add and subtract it inside
the “[ ]”. You have added ZERO to the RHS.
𝑏
𝑏
5 y = a[( x - )2
Replace the first 3 terms inside the “[ ]” by the
– (2𝑎)2 ] + c
2𝑎
square of a binomial.
𝑏
𝑏
2
2
6 y = a(x - )
Distribute the coefficient (5) to the two terms
– a (2𝑎) + c
2𝑎
inside the “[ ]”
2
𝑏
𝑏
4𝑎𝑐
7 y = a(x - )2 –
Simplify the term on the right.
+
2𝑎
𝑏
y = a(x - 2𝑎
8
9
y +
𝑏 2 −4𝑎𝑐
y- -
4𝑎
4𝑎
𝑏 2 −4𝑎𝑐
–
4𝑎
𝑏
Move the constant term on the right to the LHS.
= a(x - 2𝑎 )2
4𝑎
𝑏 2 −4𝑎𝑐
4𝑎
)2
𝑏
= a(x - 2𝑎 )2 = ANSWER
10 Vertex = (- 𝑏 , 2𝑎
Steepness = a.
𝑏 2 −4𝑎𝑐
4𝑎
)
Rewrite in vertex form (need “-“ signs).
Find the vertex and the steepness