PRESENTED BYAKILI THOMAS, DANA STA. ANA,

& MICHAEL BRISCO

Graphing Quadratic funtions in Standard Form

Section 4.1

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.

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))=2Then find the y- coordinate of the vertex.             Y= 2(2)-8(2)+6=-2So the vertex is (2,-2).Plot this point.

-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).  

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

Section 4.3 Solving x2+bx+c=0 by factoringExampleSolve 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.

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)

EXAMPLE 1 Use properties of square roots

Simplify the expression.

2=    61. 72 236=

=   2

6

2. 4 6 24= 4 6=

GUIDED PRACTICEGUIDED PRACTICE Use properties of square roots

4 12

16

144

49121

7

11

(√16÷√144) =

(√49 ÷ √121) =

Rationalizing the DenominatorForm of the

denominatorMultiply numerator and

denominator by:

√b √b

a + √b a - √b

a - √b a + √b

EXAMPLE 2 Rationalize denominators of fractions

1. 5

2

=

=5

2

=5

2

2

2

10

2

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

Solving Quadratic Equations

p² + 6 = 127 3x² + 9 = 117- 6 = - 6 - 9 = - 9p² =√121 3x² = 108 ÷ 3p = ± 11 x² = √36x = ± 6

Ex.110- (6 +7i)+ 4i10-6-7i+4i4-3i

• First, simplify the expression• Then, grouped the like terms together• Finally, write the answer in the correct form

Ex. 1(9-2i)(-4+7i)-36+63i +8i-14i²-36+71i-14(-1)-36+71i+14-22+71i

First, multiply using FOILSecondly, turn i²= -1Then, simplify, combine like termsFinally, write the answer in the standard form

Distributive Property: (2 + 3i) • (4 + 5i)= 2(4 + 5i) + 3i(4 + 5i= 8 + 10i + 12i + 15= 8 + 22i + 1 = 8 + 22i -1 = -7 + 22i  

 Be sure to replace i2 with(-1) and proceed with the simplification. 

Answer should be in a + bi form.

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.

Completing the square

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

• The three types on how to write a quadratic equation.

1.Vertex Form2.Intercept Form3.Standard Form

• Use vertex form when the vertex is given.

y= a(x-h)²+k

• Use the intercept form when x-intercepts are given.

y= a(x-p)(x-q)

• Use the standard form when 3 coordinates are given.

(-2,-1) (1,2) (3, -6)