1. LearnZillion Notes:

2. LearnZillion Notes:

3. LearnZillion Notes:
You already know that the function f(x) = x2 is a parabola with the origin as its vertex.

4. f(x)= x2 +k is a vertical translation of f(x)= x2
by k units.
LearnZillion Notes:
You also know that a function in the form f(x)= x2 +k is a vertical translation of the function f(x)= x2 by k units.
For example, f(x)= x^2+3 is a vertical translation of the function x^2 of 3 units in the positive y direction. This means that all the points, including the vertex, are vertically translated by positive 3 units.

5. f(x)= (x-h)2 is a horizontal translation of f(x)=x2
by h units.
LearnZillion Notes:
You also know that a function in the form f(x)= (x-h)2 is a horizontal translation of the function f(x)= x2 by h units.
For example, f(x)= (x-2)^2 is a horizontal translation of the function x^2 of 2 units in the positive x direction. This means that all the points, including the vertex, are horizontally translated by positive 2 units.

6. f(x) = x2 -8x + 21 f(x) = (x2-8x +16) +21 -16 f(x) = (x-4)2 + 5
LearnZillion Notes:
You also know how to take an equation written in standard form and rewrite it by completing the square.
For example, f(x) = x2 -8x can be rewritten as f(x) = (x-4)2 + 5
This form of the a quadratic equation is called the vertex form. In general vertex form is written as f(x) = (x-h)2 + k
We’ll talk a little more about why it is called vertex form in just a minute.

7. f(x)= x2 -6x + 13 f(x)= (x-3)2 + 4 LearnZillion Notes:
Let’s look at the function f(x)= x2 -6x Can you look at this function written in standard form and tell exactly where it’s vertex is located on a coordinate plane? It is extremely difficult to do so.
This function can be re-written in vertex form as f(x)= (x-3) Why is this called the vertex form? Let’s look at what we know. We know that the x-3 inside the parentheses means the vertex translates positive 3 units horizontally from the origin. We also know that the constant term of +4 means the vertex translates positive 4 units. When we combine this into one translation from the origin, we get a new vertex of (3, 4)
Take a look at the graph of this function and then look back at the function written in vertex form. We can actually see the vertex revealed in the structure of the equation. Let’s look at one more example to see if you can predict the location of the vertex.

8. f(x)= x2 +4x + 1 f(x)= (x+2)2 -3 LearnZillion Notes:
Let’s look at the function f(x)= x2 +4x Can you look at this function written in standard form and tell exactly where it’s vertex is located on a coordinate plane?
Now try it written in vertex form: f(x)= (x+2) Can you see the location of the vertex? The x+2 in parentheses tells us we have a horizontal translation of -2 and the constant term of -3 tells us that we have a vertical translation of -3 so the new vertex is (-2, -3)
Again, looking at the graph of this function we can confirm that (-2,-3) is indeed the vertex of this function.

9. For example, the vertex of
f(x) = (x-h)2 + k
For example, the vertex of
f(x) = (x-4)2 + 3
must be (-4, 3)
Actual vertex: (4, 3)
LearnZillion Notes:
A common mistake is to think that the vertex has the same sign as h in the vertex form rather than the opposite sign of h in f(x) = (x-h)2 + k
For example, it would be easy to think the vertex of f(x) = (x-4) is (-4, 3). This is a mistake, however, since the actual vertex is (4,3).

10. LearnZillion Notes: --

11. Solution: f(x) = (x+3)2 -5 vertex: (-3, -5) LearnZillion Notes:
The leading coefficient is positive so the function will have a minimum value.
We can confirm this is true by taking a quick look at the graph of the function.

12. Solution: f(x) = 2(x-1)2 +3 vertex: (1, 3) LearnZillion Notes:
The leading coefficient is positive so the function will have a minimum value.
We can confirm this is true by taking a quick look at the graph of the function.

13. LearnZillion Notes:

14. LearnZillion Notes:
--”Quick Quiz” is an easy way to check for student understanding at the end of a lesson. On this slide, you’ll simply display 2 problems that are similar to the previous examples. That’s it! You won’t be recording a video of this slide and when teachers download the slides, they’ll direct their students through the example on their own so you don’t need to show an answer to the question.