The document outlines the process of translating quadratic functions from standard form (y = ax² + bx + c) to vertex form (y = a(x - h)² + k) when a = 1 using the completing the square method. It includes step-by-step examples, starting with transposing the constant term, completing the square, simplifying the equation, and finally rewriting it in vertex form. Multiple examples demonstrate the process applied to different quadratic functions.

1. Translating Quadratic
Function from Standard Form
into Vertex Form if 𝒂 = 𝟏
Mathematics 9

2. Standard From and Vertex Form of
Quadratic Function
Standard Form
𝑦 = 𝑎𝑥2
+ 𝑏𝑥 + 𝑐
Vertex Form
𝑦 = 𝑎 𝑥 − ℎ 2
+ 𝑘

3. To translate quadratic function
from standard form to vertex
form, you need to know the
following:
1. Completing the Square Method
2. Factoring

4. Standard to Vertex Form
 Steps for translating quadratic function from standard to
vertex form if 𝒂 = 𝟏
Step 1: Since the equation is in the standard form 𝑦 = 𝑎𝑥2 +
𝑏𝑥 + 𝑐, and we want to convert it into the form of 𝑦 =
𝑎 𝑥 − ℎ 2 + 𝑘, then the first thing that we need to do is
transpose 𝑐 to the other side of equal sign.
Example 1:
𝑦 = 𝑥2 + 4𝑥 + 7
𝑦 − 7 = 𝑥2
+ 4𝑥

5. Standard to Vertex Form
 Steps for translating quadratic function from standard to
vertex form 𝒂 = 𝟏
Step 2: Perform completing the square. The goal of this method is to make a
perfect square trinomial and it will only happen if the coefficient of 𝑥2
or the
value of a is equal to 1. Since the coefficient of 𝑥2
in the example below is
equal to 1, then we can immediately perform completing the square. For this
situation, we will going to add
𝒃
𝟐
𝟐
to the both sides of equation.
𝑦 − 7 +
𝒃
𝟐
𝟐
= 𝑥2 + 4𝑥 +
𝒃
𝟐
𝟐
𝒚 − 𝟕 + 𝟒 = 𝒙 𝟐
+ 𝟒𝒙 + 𝟒
Since 𝑏 = 4, then
𝟒
𝟐
𝟐
= 4

6. Standard to Vertex Form
 Steps for translating quadratic function from standard to
vertex form 𝒂 = 𝟏
Step 3: Simplify both sides of equation.
𝒚 − 𝟕 + 𝟒 = 𝒙 𝟐
+ 𝟒𝒙 + 𝟒
To simplify, add -7
and 4
Since this is already a
perfect square trinomial,
then rewrite it as square of
binomial:
𝑥 +
𝑏
2
2
𝒚 − 𝟑 = 𝒙 + 𝟐 𝟐

7. Standard to Vertex Form
 Steps for translating quadratic function from standard to
vertex form 𝒂 = 𝟏
Step 4: Transpose the constant term to the other side of the
equal sign so that ONLY 𝒚 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏 will be left.
𝒚 − 𝟑 = 𝒙 + 𝟐 𝟐 Since the constant is -3,
when you transpose it, the
sign will change.
𝒚 = 𝒙 + 𝟐 𝟐
+ 𝟑
This is already the vertex
form of the equation
𝒚 = 𝒙 𝟐 + 𝟒𝒙 + 𝟕
Final Answer

8. More Examples
Translating Standard Form into Vertex Form when 𝑎 = 1

9. Example 1
𝒚 = 𝒙 𝟐
+ 𝟔𝒙 + 𝟏𝟎 Quadratic in Standard Form
𝒚 − 𝟏𝟎 = 𝒙 𝟐
+ 𝟔𝒙 Transpose 10 to the left
𝒚 − 𝟏𝟎 + _____ = 𝒙 𝟐 + 𝟔𝒙 + _____
Completing the square:
Since 𝑏 = 6, the
𝑏
2
2
=
6
2
2
= 9
𝒚 − 𝟏𝟎 + 𝟗 = 𝒙 𝟐 + 𝟔𝒙 + 𝟗 Add 9 to both sides of equation
𝒚 − 𝟏 = 𝒙 + 𝟑 𝟐 Transpose -1 so that only y will be
left.
𝒚 = 𝒙 + 𝟑 𝟐
+ 𝟏 Final Answer
Find the vertex form of the function 𝑦 = 𝑥2
+ 6𝑥 + 10.

10. Example 2
𝒚 = 𝒙 𝟐
+ 𝟐𝒙 − 𝟑 Quadratic in Standard Form
𝒚 + 𝟑 = 𝒙 𝟐
+ 𝟐𝒙 Transpose -3 to the left
𝒚 + 𝟑 + _____ = 𝒙 𝟐 + 𝟐𝒙 + _____
Completing the square:
Since 𝑏 = 2, the
𝑏
2
2
=
2
2
2
= 1
𝒚 + 𝟑 + 𝟏 = 𝒙 𝟐 + 𝟐𝒙 + 𝟏 Add 1 to both sides of equation
𝒚 + 𝟒 = 𝒙 + 𝟏 𝟐 Transpose 4 so that only y will be
left.
𝒚 = 𝒙 + 𝟏 𝟐
− 𝟒 Final Answer
Find the vertex form of the function 𝑦 = 𝑥2
+ 2𝑥 − 3.