Completing the Square with Parabolas

Perfect Square Trinomials Examples x2 + 6x + 9 x2 - 10x + 25 x2 + 12x + 36

Creating a Perfect Square Trinomial In the following perfect square trinomial, the constant term is missing. X2 + 14x + ____ Find the constant term by squaring half the coefficient of the linear term. (14/2)2 X2 + 14x + 49

Perfect Square Trinomials Create perfect square trinomials. x2 + 20x + ___ x2 - 4x + ___ x2 + 5x + ___ 100 4 25/4

Completing the Square with Parabolas Example 1: Put the following parabola in vertex form by completing the square: Step 1: Move the constant, c, to the other side. 𝑦= 𝑥 2 +8𝑥−20 +20 +20 𝑦+20= 𝑥 2 +8𝑥

Completing the Square with Parabolas Step 2: Find 𝑏 2 . 𝑦+20= 𝑥 2 +8𝑥 𝑏 2 = 8 2 =4

Completing the Square with Parabolas Step 3: Take 𝑏 2 2 𝑏 2 = 8 2 =4 4 2 =16

Completing the Square with Parabolas Step 4: Add 𝑏 2 2 to both sides of the equation. 𝑦+20= 𝑥 2 +8𝑥 +16 +16

Completing the Square with Parabolas 𝑦+36=𝑥 2 +8𝑥+16 𝑦+36= 𝑥+4 2 Step 5: Factor the perfect square trinomial.

Completing the Square with Parabolas 𝑦+36= 𝑥+4 2 𝑦= 𝑥+4 2 −36 Step 6: Move the constant back to the right side. −36 −36 Can you identify the vertex of this parabola?

Completing the Square with Parabolas Example 2: Put the following parabola in vertex form by completing the square: Step 1: Move the constant, c, to the other side. 𝑦= 𝑥 2 −4𝑥+1 −1 −1 𝑦−1= 𝑥 2 −4𝑥

Completing the Square with Parabolas Step 2: Find 𝑏 2 . 𝑦−1= 𝑥 2 −4𝑥 𝑏 2 = −4 2 =−2

Completing the Square with Parabolas Step 3: Take 𝑏 2 2 𝑏 2 = −4 2 =−2 (−2) 2 =4

Completing the Square with Parabolas Step 4: Add 𝑏 2 2 to both sides of the equation. 𝑦−1= 𝑥 2 −4𝑥 +4 +4

Completing the Square with Parabolas 𝑦+3=𝑥 2 −4𝑥+4 𝑦+3= 𝑥−2 2 Step 5: Factor the perfect square trinomial.

Completing the Square with Parabolas 𝑦+3= 𝑥−2 2 𝑦= 𝑥−2 2 −3 Step 6: Move the constant back to the right side. −3 −3 Can you identify the vertex of this parabola?

Completing the Square with Parabolas Example 3: Put the following parabola in vertex form by completing the square: Step 1: Move the constant, c, to the other side. 𝑥= 𝑦 2 +12𝑦+25 −25 −25 𝑥−25= 𝑦 2 +12𝑦

Completing the Square with Parabolas Step 2: Find 𝑏 2 . 𝑥−25= 𝑦 2 +12𝑦 𝑏 2 = 12 2 =6

Completing the Square with Parabolas Step 3: Take 𝑏 2 2 𝑏 2 = 12 2 =6 6 2 =36

Completing the Square with Parabolas Step 4: Add 𝑏 2 2 to both sides of the equation. 𝑥−25= 𝑦 2 +12𝑦 +36 +36

Completing the Square with Parabolas 𝑥+11=𝑦 2 +12𝑦+36 𝑥+11= 𝑦+6 2 Step 5: Factor the perfect square trinomial.

Completing the Square with Parabolas 𝑥+11= 𝑦+6 2 𝑥= 𝑦+4 2 −11 Step 6: Move the constant back to the right side. −11 −11 Can you identify the vertex of this parabola?

Completing the Square with Parabolas Example 4: Put the following parabola in vertex form by completing the square: Step 1: Move the constant, c, to the other side. 𝑦=4 𝑥 2 −24𝑥−5 +5 +5 𝑦+5= 4𝑥 2 −24𝑥

Completing the Square with Parabolas Step 2: Find 𝑏 2 . **In order to complete the square, 𝑎=1. **Before we can find 𝑏 2 , we must factor out the 𝑎=4. 𝑦+5= 4𝑥 2 −24𝑥 𝑦+5=4( 𝑥 2 −6𝑥) 𝑏 2 = −6 2 =−3

Completing the Square with Parabolas Step 3: Take 𝑏 2 2 𝑏 2 = −6 2 =−3 (−3) 2 =9

Completing the Square with Parabolas Step 4: Add 𝑏 2 2 to both sides of the equation. 𝑦+5= 4(𝑥 2 −6𝑥 +36 +9)

Completing the Square with Parabolas 𝑦+41=4(𝑥 2 −6𝑥+9) 𝑦+41= 4 𝑥−3 2 Step 5: Factor the perfect square trinomial.

Completing the Square with Parabolas 𝑦+41= 4 𝑥−3 2 𝑦= 4 𝑥−3 2 −41 Step 6: Move the constant back to the right side. −41 −41 Can you identify the vertex of this parabola?

Completing the Square with Parabolas 𝑥=2 𝑦 2 −28𝑦+10 You Try! Put the following parabola in vertex form by completing the square: Can you identify the vertex of this parabola?