1. Quadratic Functions

2. p(x) = an xn + an-1 xn-1 + … + a1 x + a0
PolyNomials
Definition:
A polynomial function is a function that can be expressed in the form:
p(x) = an xn + an-1 xn-1 + … + a1 x + a0
Where an , an-1 , … , a1 , a0 are real numbers, an ≠ 0, the exponents are non-negative integers ✔
The degree is 2 ✔
The degree is 1 ✔
The degree is 0 ✔
The degree is 3
Definition:
The degree of a polynomial is largest exponent of x.

3. Quadratic functions Definition: p(x) = ax2 + bx + c
A polynomial of degree 0 is called a constant function.
A polynomial of degree 1 is called a linear function.
Definition:
A degree 2 polynomial function is called a quadratic function. The general form a quadratic function is
p(x) = ax2 + bx + c
where a, b, and c are real numbers with a ≠ 0.
Quadratic functions are incredibly important functions that show up everywhere in the real world.

4. Parabolas p(x) = ax2 + bx + c a > 0 a < 0
The graph of a quadratic polynomial is called a parabola.
p(x) = ax2 + bx + c
Axis of
Symmetry
vertex
Axis of
Symmetry
vertex
a > 0
a < 0

5. Parabolas a decreases from 1 towards 0
How does the graph of a quadratic function change as we change a, b, and c?
a decreases from 1 towards 0

6. Parabolas a increases from 1 to 10
How does the graph of a quadratic function change as we change a, b, and c?
a increases from 1 to 10

7. Parabolas c increases from 0 to 2
How does the graph of a quadratic function change as we change a, b, and c?
c increases from 0 to 2

8. Parabolas c decreases from 0 to -2
How does the graph of a quadratic function change as we change a, b, and c?
c decreases from 0 to -2

9. Standard form Definition: p(x) = a(x – h)2 + k
The standard form of a quadratic function is
p(x) = a(x – h)2 + k
Where (h, k) is the vertex of its graph and a ≠ 0.

10. Summary General Form: standard Form: Vertex: Axis of symmetry: Vertex:
Parabola opens up
Parabola opens down

11. problems
Find the vertex and the x-intercepts of the following functions:

12. problems
Find the quadratic function with the indicated vertex that passing though the given point:
1. Vertex: (2,3)
Point: (0,2)
2. Vertex: (-2,-2)
Point: (-1,0)
3. Vertex: (6,6)
Point: (1/2, 3/4)

13. problems
The profit P (in hundreds of dollars) that a company makes depends on the amount x (in hundreds of dollars) that the company spends on advertising according to the model:
P(x) = x – 0.5x2
How much should the company spend on advertising to maximize profits?