Name:
Date:
Algebra 2
Putting Functions into Vertex Form
As a reminder, here’s a quadratic function in vertex form: f(x) = a(x – h)2 + k. The vertex is at
(h, k), and the a value tells the parabola’s direction ( if a > 0,  if a < 0).
How to put f(x) = ax2+bx+c into vertex form
Method 1: Completing the Square
Put the function f(x) = 3x2 – 18x + 23 into vertex form, and identify the vertex.
First move the c value to the other side just like you do when equation solving
f(x) – 23 = 3x2 – 18x
Next, factor out the 3 from the x2 and x terms:
f(x) – 23 = 3(x2 – 6x)
Think about how to complete the perfect square: x2 – 6x + ? . As before, the needed
number is found by halving and squaring: x2 – 6x + __9__. So we want to insert a 9 inside the
parentheses.
Now here’s the crucial step: inserting a 9 would increase the function’s value by 9·3=27.
So, we need to offset this increase by adding 27 to the other side.
f(x) – 23 + 27 = 3(x2 – 6x + 9)
Write the perfect square as a square, and combine the – 23 +27 to get:
f(x) + 4 = 3 (x – 3)2.
Finally, subtract the four from both sides to get:
f(x) = 3 (x – 3)2 – 4
vertex: (3, –4)
Method 2: Find the Vertex
Put the function f(x) = –2x2 – 20x + 10 into vertex form, and identify the vertex.
Find the vertex by using our formula x =
x=
-b
2a
-(-20) 20
=
= -5
2(-2) -4
This is the x-coordinate. Plug into the function to find y:
f(-5) = (-2)(-5) 2 – (20)(-5) +10 = -50 +100 + 10 = 60
Plug the vertex into vertex form, keeping the a value from standard form:
f(x) = –2(x + 5)2 + 60
vertex: (–5, 60)
Name:
Date:
Algebra 2
You try it
Put the function f(x) = 4x2 + 32x + 50 into vertex form, and identify the vertex. Use both
methods.
Name:
Date:
Problem Set
1. Put each of these functions into vertex form, and identify the vertex.
a. f(x) = 2x2 + 4x – 1
b. f(x) = 5x2 – 20x + 17
c. f(x) = 4x2 + 24x + 36
d. f(x) = 3x2 + 36x
Algebra 2
Name:
Date:
e.
Algebra 2
f(x) = –2x2 – 12x – 15
f. f(x) = –3x2 + 12x – 20
g. f(x) = –x2 + 8x + 5
2. Find the vertex of each of these functions on your calculator. These are the same functions
that were in problem 1, so you should get answers that agree with that problem. (If not, check
your work in problem 1 for errors.)
a. f(x) = 2x2 + 4x – 1
vertex:
b. f(x) = 5x2 – 20x + 17
vertex:
c. f(x) = 4x2 + 24x + 36
vertex:
d. f(x) = 3x2 + 36x
vertex:
f(x) = –2x2 – 12x – 15
vertex:
f. f(x) = –3x + 12x – 20
vertex:
g. f(x) = –x + 8x + 5
vertex:
e.
2
2
Name:
Date:
Algebra 2
3. Put each of these functions into vertex form, then make a rough sketch of the graph. Your graph
should show the correct shape ( vs. ) and the vertex should be labeled with coordinates.
a. f(x) = 2x2 + 4x – 3
b. f(x) = –3x2 + 12x – 10
c. f(x) = –x2 + 6x – 9