1. Family of Quadratic Functions
Lesson 4-2b

2. General Form
Quadratic functions have the standard form y = ax2 + bx + c
a, b, and c are constants
a ≠ (why?)
Quadratic functions graph as a parabola

3. Zeros of the Quadratic Zeros are where the function crosses the x-axis
Where y = 0
Consider possible numbers of zeros 
None (or two complex)
One
Two

4. Axis of Symmetry Parabolas are symmetric about a vertical axis
For y = ax2 + bx + c the axis of symmetry is at
Given y = 3x2 + 8x
What is the axis of symmetry?

5. Vertex of the Parabola The vertex is the “point” of the parabola
The minimum value
Can also be a maximum
What is the x-value of the vertex?
How can we find the y-value?

6. Vertex of the Parabola Given f(x) = x2 + 2x – 8
What is the x-value of the vertex?
What is the y-value of the vertex?
f(-1)= 1 – 2 – 8 = -9
The vertex is at (-1, -9)

7. Vertex of the Parabola Given f(x) = x2 + 2x – 8
Graph shows vertex at (-1, -9)
Note calculator’s ability to find vertex (minimum or maximum)

8. Shifting and Stretching
Start with f(x) = x2
Determine the results of transformations
___ f(x + a) = x2 + 2ax + a2
___ f(x) + a = x2 + a
___ a * f(x) = ax2
___ f(a*x) = a2x2
a) horizontal shift
b) vertical stretch or
squeeze
c) horizontal stretch or
d) vertical shift
e) none of these

9. Experiment with Quadratic Function Spreadsheet
Other Quadratic Forms
Standard form y = ax2 + bx + c
Vertex form y = a (x – h)2 + k
Then (h,k) is the vertex
Given f(x) = x2 + 2x – 8
Change to vertex form
Hint, use completing the square
Experiment with Quadratic Function Spreadsheet

11. Vertex Form Changing to vertex form
Add something in to make a perfect square trinomial
Changing to vertex form
Subtract the same amount to keep it even.
This gives us the ordered pair (h,k)
Now create a binomial squared

12. Write the following in Vertex Form
y = 𝒙 𝟐 +𝟖𝒙+𝟒

13. Vertex Form Changing to vertex form y = -3 𝒙 𝟐 + 6x - 1
Add something in to make a perfect square trinomial
Changing to vertex form
Factor a from the linear and the quadratic terms
y = -3 𝒙 𝟐 + 6x - 1
y = -3 (𝒙 𝟐 - 2x ) – 1+
y = -3 (𝒙 𝟐 - 2x ) – 1 + 3
Subtract the same amount to keep it even.
y = -3 (𝒙−𝟏) 𝟐 – 4
This gives us the ordered pair (h,k) = (1, -4)
Now factor to create a binomial squared

14. Write the following in Vertex Form
y = −𝟒𝒙 𝟐 +𝟏𝟔𝒙 −𝟓

15. Write the following in Vertex Form
y = 2 𝒙 𝟐 +𝟏𝟎𝒙+𝟕

16. For each function, the vertex of the function’s graph is given
For each function, the vertex of the function’s graph is given. Find the unknown coefficients.
y = -3 𝑥 2 +𝑏𝑥+𝑐, given (1,0)

17. y = 𝑐 - a 𝑥 2 - 2x, given vertex (-1,3)
For each function, the vertex of the function’s graph is given. Find the unknown coefficients.
y = 𝑐 - a 𝑥 2 - 2x, given vertex (-1,3)

18. Assignment Book Page 206: 26-31, 39-43odd, 44-47, 49-52