quadratic function lesson using doceri

1. Quadratic Functions
Graphing and Modeling

2. Quadratic Functions as Projectiles
f (x)= ax2
+bx+c where a, b and c are real and a ¹ 0.

3. Characteristics
of Graphs

4. Relating Solutions of the Quadratic
Equation with x-intercepts

5. Relating Graphs with the
Quadratic Formula

6. The Basic Quadratic and the
Transformed Quadratic

7. Graphing Quadratic Function
in Vertex Form – The Steps
Determine whether the
parabola opens UP or
DOWN
Determine the vertex
(h,k)
Find any x-intercepts
by solving f(x)=0
Find the y-intercept
by computing f(0)
Plot the intercepts
and vertex
f (x) = a(x -h)2
+k

8. Graphing Quadratic Function
in Vertex Form – An Example
f (x) = -2(x -3)2
+8
a = -2 h = 3 k = 8
Since a is negative
we know the parabola
opens DOWN
Since the vertex has
the form (h, k), our
vertex will be (3, 8)

9. Graphing Quadratic Function in
Vertex Form – An Example
Solving for the x- and y-intercepts

10. Graphing Quadratic Function in
Vertex Form – An Example

11. Graphing Quadratic Function
in General Form – The Steps
Determine whether the
parabola opens UP or
DOWN
Determine the vertex
Find any x-intercepts
by solving f(x)=0
Find the y-intercept
by computing f(0)
Plot the intercepts
and vertex
f (x) = ax2
+bx+c
-
b
2a
, f -
b
2a
æ
è
ç
ö
ø
÷
æ
è
ç
ö
ø
÷

12. Graphing Quadratic Function in
General Form – An Example
f (x) = x2
+4x+1
a =1
b = 4
Since a is positive
we know the
parabola opens UP
x-coordinate
of the vertex:
y-coordinate
of the vertex:
The vertex:
-2,-3( )

13. Graphing Quadratic Function in
General Form – An Example
Solving for the x- and y-intercepts

14. Graphing Quadratic Function in
General Form – An Example

15. The Parabolic Path of a
Punted Football
When a football is kicked, the height of
the punted football, f(x), in feet, can
be modeled by
f (x)= -0.01x2
+1.18x+2
where x is the ball’s horizontal
distance, in feet, from the point of impact
with the kicker’s foot.
a. What is the maximum height of the punt?
x = -
b
2a
= -
1.18
2(-0.01)
= -(-59) = 59 feet

16. The Parabolic Path of a
Punted Football
When a football is kicked, the height of
the punted football, f(x), in feet, can
be modeled by
f (x)= -0.01x2
+1.18x+2
where x is the ball’s horizontal
distance, in feet, from the point of impact
with the kicker’s foot.
a. What is the maximum height of the punt?
f (59)= -0.01(59)2
+1.18(59)+2
The maximum height of the punt occurs 59 feet from the
kicker’s point of impact. The actual maximum height of the
punt is
=36.81 feet

17. The Parabolic Path of a Punted
Football Continued
f (x)= -0.01x2
+1.18x+2
b. How far must the nearest defensive
player, who is 6 feet from the
kicker’s point of impact, reach to
block the punt?
This means we need to find the height
of the ball 6 feet from the kicker.
In other words, “plug in” 6 for x.
f (6)= -0.01(6)2
+1.18(6)+2 =8.72 feet
The defensive player must reach 8.72
feet above the ground to block the punt.

18. Key Points to Know for Graphing
Quadratic Functions:
General form versus Vertex form
Understanding the shape of a quadratic
function
When a parabola opens up or down
Using either form to graph a parabola
Able to solve for a maximum or minimum
Able to solve for x- and y-intercepts

19. I Challenge You…
Write a quadratic function in standard
form that models the area of the shaded
region.
x +9
x+5
x +1 x+3
x +3
x -1
x
x