1. 3.1 Exploring Quadratic Relations
“Pop Fly” in Baseball
-1As
Hard to catch, since it’s falling straight down
It’s modeled by the relation:
y = -5x2 + 20x + 1
x: time in seconds after ball leaves the bat
Height equation is an example of a quadratic relation in standard form
PARABOLA ->

2. Standard Form
Quadratic Relation in Standard Form is a relation in the form:
y = ax2+ bx + c, where a ≠0
In the baseball’s height relation:
y = -5x2 + 20x + 1
a = -5; b = 20; c = 1

3. Explore Changes to Standard Form
How does changing the coefficients and constant in y = ax2 + bx + c affect the graph of the quadratic relation?

4. Analyzing Further y = -5x2 + 20x + 1 – Make a table of values x y
1st Diff
2nd Diff -3
-104 - -2
-59 45 -1
-24 35
-10 1 25 16 15 2 21 5 3 -5 4
-15
-25
2nd differences are the same, so we can conclude that we have a quadratic relationship
A parabola is a graph of a symmetric quadratic relation

5. What do they have in common?
y = -5x2 + 20x + 1
y = x2 - 50x + 6
y = -2x2 + 2x - 3
y = 4x2 - x - 1
The highest degree is always 2for a quadratic relationship.
Which one of these is not a quadratic relationship?
a) y = 2x2 + 3
b) y = 4x - 9
c) y = -x2 + x
d) y = x2 + 3x + 9
The highest degree in (b) is 1, not 2

6. In Summary… Graph of a quadratic relation: y = ax2+bx+c, where a≠0
Any relation with a polynomial of degree 2 is quadratic
2nd differences must be constant, not zero
When a > 0, parabola opens up
When a < 0, parabola opens down
Changing b changes location of line of symmetry
Constant c is the y-intercept of the parabola