PRESENTED BY AKILI 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))=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.

-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=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.

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.

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. 1. 72 = 36 2 =    6 2 6 2. 4 6 = 24 = 4 6 =   2

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

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

EXAMPLE 2 Rationalize denominators of fractions 5 5 1. = 2 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 = - 9 p² =√121 3x² = 108 ÷ 3 p = ± 11 x² = √36 x = ± 6

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

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

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                                       36 X + 3 = ± 6                  3. Simplify X= 3 ± 6                      4. Isolate the x. The solutions are x = 9 and x = -3

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