1. Analysis of Linear and quadratic polynomials
L(x) = ax + b, x a variable; a, b unspecified constants
Basic question: determine when L(x) < 0 , L(x) = 0, L(x) >0
In terms of a and b
We can work this out geometrically, using the x-intercept and
slope of L(x):

2. Question: For what values of x is L(x) = 3*x – 8 positive?
Question: Suppose a line is given parameterically:
x = 5t + 2, y = -3t + 4
For what values of x is y positive?

3. Question: Suppose L(x) = 4x – 3hx + d.
For what values of x is L(x) < 0?

4. Problem: The weight w(t) of an object is a linear polynomial
in t. It is found by experiment that w(4) = 29 and w(7) = 29 – 1/10.
For what values of t is w(t) >0?

5. Quadratic case:
Q(x) = ax^2 + bx + c
For what values of x is Q(x) positive? For what values of x is Q(x)
negative?
Case 1:
If Q(x) factors into a(x-u)(x-v) we can easily answer it:
Need this basic fact: The product of a bunch of numbers is negative
When an ____ number of them are negative.

6. Problem: For what values of x is Q(x)= -3/7(x-8)(x+5) negative?

7. Use of completing the square to handle the general case.
Completing the square on x^2 + cx

8. Example: For what values of x is x^2 + 4x – 7 positive?

9. Extreme points of quadratics: a number p is called an extreme point of
the quadratic Q(x) = ax^2 + bx + c if Q(p) >= Q(x) for all x
or Q(p) <= Q(x) for all x. In the first case, Q(p) is a maximum value of Q(x)
In the second case, Q(p) is minimum value.

10. Theorem: For the quadratic Q(x) = ax^2 + bx + c, the number p = -b/(2a)
Is the extreme point of Q(x). Furthermore, Q(p) is a maximum if a < 0
and a minimum if a > 0.

11. Application: A ball is thrown into the air by a 5 foot thrower.
At time t, its height above the ground is h(t) = -16t^2 + 50t + 5.
when does it get to its highest point, how high is it then, and
when does it hit the ground ?

12. Problem:
A quadratic Q(x) has an extreme Q(3) = 5 and a root Q(7) = 0.
Find Q(0).