1. Section 3.6 Quadratic Equations Objectives
1. Find the vertex and standard equation of a parabola
2. Sketch the graph of a parabola
3. Solve applied problem involving maximum or minimum
Best way to view this presentation is thru power point XP

2. Definition of a quadratic Function: A function f is a quadratic function if
f(x) = ax2 + bx + c
Where a, b and c are constants with a ≠ 0
Example 1. Sketch the graph of f if
f( x ) = x2
Note: b = c = 0
Solution:
Table of variation
(x,y) y x
(-2, -2) -2
(-1, - ½ )
- ½ -1
(0, 0 )
(1, -1/2 )
-1/2 1
(2, - 2 )
- 2 2

3. Note: Only b = 0
Solution:
It is enough to shift the previous graph by 4 units upward as we can see below

4. Expressing a Quadratic Equation as f(x) = a ( x – h )2 + k
Method By completing the square of the right hand side of the quadratic equation and it will be explained in class.
Method 2. ( Zalzali’s Method )
Step 1. Let h =
Step 2. Let k =
Step 3. Insert the values of h and k in f ( x ) = a( x – h )2 + k

5. Then f(x) = 3 ( x + 4 )2 +2 Class Work 1
Example 3. Write the following quadratic function in the standard form f(x) = a ( x –h )2 + k
Solution:
Then f(x) = 3 ( x + 4 )2 +2
Class Work 1
Write the following quadratic function f(x) = -x2 -2x +8
in the standard form
f(x) = a ( x –h )2 + k
Answer: f(x) = - ( x + 1 )2 +9

6. Standard Equation of a parabola with vertical Axis
The graph of the equation
f(x) = a ( x –h )2 + k
For a ≠ 0 is a parabola with vertex V(h, k ) and it has a vertical axis x = h.
The parabola opens upward if a > and
The parabola opens downward if a < 0.
Notes about Vertex V ( h , k )
Note 1. If a < 0, then the vertex V ( h, k ) is a maximum point of the parabola
Note 2. If a > 0, then the vertex V ( h, k ) is a minimum point of the parabola

7. Graphing Parabolas Solution: Strategy: 1. Find the vertex V ( h, k )
2. Identify if the vertex is maximum or minimum
3. Find x and y intercepts if they exist
4. Plot the vertex and the intercepts
5. Plot the vertical axis ( it is also called axis of symmetry of a parabola )
6. Connect the points of the parabola and extend the graph
Example 4: Graph the parabola
Solution:
1. Vertex V ( -1, 9) Check class practice
f(x) = - x2 – 2x + 9
2. a = -1 < 0 ( Parabola is open downward )
and has a maximum at the vertex
3. x-intercept(s): Set y = 0, then x =-4 and 2 Therefore, points are ( -4, 0 ) and ( 2, 0 )
y-intercept : Set x = 0 , then y = 8. Point ( 0, 8 )
x = -1

8. Class Work 2
Graph the parabola

9. Word Problem
Height of a projectile: An object is projected vertically upward from the top of a building with an initial velocity of 144 ft / sec. Its distance s( t ) = - 16t t + 100
(a) Find its maximum distance above the ground
(b) Find the height of the building.
Solution:
Height =s(0)

10. Do all homework problems in the syllabus
Homework of section 3.6
Do all homework problems in the syllabus