1. PRESENTED BY AKILI THOMAS, DANA STA. ANA, & MICHAEL BRISCO

2. Graphing Quadratic funtions in Standard Form
Section 4.1

3. Graph Quadratic Functions in Standard Form 4.1
A quadratic function is a function that can be written in the standard form y = ax2+bx+c where a doesn’t equal 0. The graph of a quadratic function is a parabola.

4. Graph a function of the form y= ax2+bx+c
-Graph y= 2x2-8x+6
-Step 1 Identify the coeficients of the function. The coefficients are a=2, b=-8, and c=6. Because a is greater than 0, The parabola opens up.
-Step 2 Find the vertex. Calculate the x coordinate.
X=-b/2a=-(-8)/(2(2))=2
Then find the y- coordinate of the vertex.             
Y= 2(2)-8(2)+6=-2
So the vertex is (2,-2).Plot this point.

5. -Step 3 Draw the axis of symmetry x=2
-Step 4 Identify the y-intercept c, which is 6. Plot the point (0,6). Then reflect this point in the axis of symmetry to plot another point, (4,6).

6. -Step 5 Evaluate the function for another value of x, such as x=1.
y=2(1)-8(1)+6=0
Plot the point (1,0) and its reflection (3,0) in the axis of symmetry.
-Step 6 Draw a parabola through the plotted points.

7. Section 4.3 Solving x2+bx+c=0 by factoring
Example
Solve x2-13x-48=0.
Use factoring to solve for x.    
x2-13x-48=0                            Write original equation.
(x-16)(x+3)=0                          Factor.
x-16=0    or    x+3=0                Zero product property.
x=16    or    x=-3                      Solve for x.

9. Properties of Square Roots
Product Property = √ab = √a × √b
Example = √18 = √9 × √2 = 3√2
Quotient Property = √a÷b = (√a÷√b)
Example = √2÷25 = (√2÷√25) = (√2÷5)

10. EXAMPLE 1
Use properties of square roots
Simplify the expression. 1. 72 = 36 2
=    6 2 6 2. 4 6 = 24 = 4 6
=   2

11. GUIDED PRACTICE
GUIDED PRACTICE
Use properties of square roots 16
144
(√16÷√144) = 4 12 49
121
(√49 ÷ √121) = 7 11

12. Multiply numerator and denominator by:
Rationalizing the Denominator
Form of the
denominator
Multiply numerator and denominator by: √b √b
a + √b
a - √b
a - √b
a + √b

13. EXAMPLE 2
Rationalize denominators of fractions 5 5 1. = 2 2 5 2 = 2 2 10 2 =

14. Solving Quadratic Equations
You can use square roots to solve quadratic equations:
If s>0, then x2 = s has two real number solutions:
X = √s and x = -√s
The condensed form of these solutions is:
X =±√s

15. Solving Quadratic Equations
p² + 6 = 127 3x² + 9 = 117
- 6 = = - 9
p² =√121 3x² = 108 ÷ 3
p = ± 11 x² = √36
x = ± 6

17. Ex.1
10- (6 +7i)+ 4i
10-6-7i+4i
4-3i
First, simplify the expression
Then, grouped the like terms together
Finally, write the answer in the correct form

18. Ex. 1 (9-2i)(-4+7i) -36+63i +8i-14i² -36+71i-14(-1) -36+71i+14 -22+71i
First, multiply using FOIL
Secondly, turn i²= -1
Then, simplify, combine like terms
Finally, write the answer in the standard form

19. Distributive Property:
(2 + 3i) • (4 + 5i)
= 2(4 + 5i) + 3i(4 + 5i
= i + 12i + 15
= i + 1
 = i -1
 = i  
 Be sure to replace i2 with(-1) and proceed with the simplification. 
Answer should be in a + bi form.

21. Completing the square
In 4.5, you solved equations of the form x² = k by finding square roots. Also, you learned how to solve quadratic equations.
In 4.7, you will learn the form, x² +bx. Also, you will learn how to complete the square. You have to add (b÷2) ² to make a perfect square trinomial.

22. Completing the square X² + 6x + 9 = 36 1. Factor out the X² + 6x + 9
( x+ 3) ² = √36            2. Square out
                                      36
X + 3 = ± 6                  3. Simplify
X= 3 ± 6                      4. Isolate the x.
The solutions are x = 9 and x = -3

23. The three types on how to write a quadratic equation.
Vertex Form
Intercept Form
Standard Form

24. Use vertex form when the vertex is given.
y= a(x-h)²+k

25. Use the intercept form when x-intercepts are given.
y= a(x-p)(x-q)

26. Use the standard form when 3 coordinates are given.
(-2,-1) (1,2) (3, -6)