3/15 L11: Completing the Square

3/15 L11: Completing the Square

Don't forget to upload your notes/work to your class notebooks in Q3: week 24 when we complete Lesson 11.

HOMEWORK TONIGHT (ASSIGNED ON THE LAST SLIDE)

Copy to your notes.

The same equation in 3 forms:

• Standard $ax^2+bx+c$   

• Factored  $a\left(x-m\right)\left(x-n\right)$  

• Vertex  $a\left(x-h\right)^2+k$  

Remember Function Transformations from  $y=x^2$   here the vertex is (0,0)

•  $y=a\left(x-h\right)^2+k$                  the transformed function is in VERTEX FORM

• vertex = (h, k)

• remember to flip the sign of "h"

• h: horizontal movement

• k: vertical movement

Multiple Choice

Look at this quadratic in VERTEX FORM. Which best describes the transformation (movement from the parent function at the origin) of the quadratic function

  $\left(x+5\right)^2-1$  

• move the vertex left 5 units and down 1

• move the vertex right 5 units and down 1

• move the vertex left 1 unit and up 5

• move the vertex right 1 unit and down 5

What's the benefit of a quadratic in standard form?

  $ax^2+bx+c$  

• It's easy to find the AOS from the formula $x=-\frac{b}{2a}$  

• standard form is the only way we can factor to find the x-intercepts (soln's)

What's the benefit of a quadratic in factored form?
 $a\left(x-m\right)\left(x-n\right)$  

• It's already factored, so we can quickly find the x-intercepts (using ZPP).
  

What's the benefit to changing a quadratic in vertex form?
 $a\left(x-h\right)^2+k$  

• We can simply look at an equation in this form and "see" the vertex (h,k)
  

• Once we know the vertex, we can identify the AOS just as easily (x-value of the vertex)

Multiple Choice

Which form of a quadratic equation is best if you want to quickly "see" the vertex with no need to calculate anything?

• standard form

• factored form

• vertex form

In your notes....Exploratory Challenge: Why is it impossible to factor this expression as a perfect square trinomial?

 $x^2+8x+3?$  

• It is not a perfect square

• If you could change something to make it a perfect square, what would you change?

• If the constant term were a 16, it would be a perfect square

• Proof:  $4^{2\ }$  as the constant term and 4(2) as the linear term coefficient

• What is the difference between the changed term (16) and the original term (3)?

• That difference gives us the vertext form $\left(x+4\right)^2-13$  

How do we "change" the form of a quadratic from standard to vertex form?

The method is called:

"COMPLETING THE SQUARE"

This image can be found in your notes.

Multiple Choice

If given the expression

  $x^2+4x-5=0$  

what is the first step in "completing the square"?

• find factors of a*c that sum to be the middle term "b"

• divide all terms by 4

• add 5 to both sides

• subtract 4 from both sides

Multiple Choice

From the same equation in vertex form, what are the coordinates of the vertex of the parabola?  $\left(x-6\right)^2-9=0$  

• (6,9)

• (-6,-9)

• (-6,9)

• (6,-9)

Homework tonight:

Lesson 11: 2 problems given at the end of Lesson 11 Notes in your class notebook.