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

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

2. Graphing Quadratic funtions in Standard Form Section 4.1

3. 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 Quadratic Functions in Standard Form 4.1

4. -Graphy= 2x2-8x+6 -Step 1Identify 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 2Find 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. Graph a function of the form y= ax2+bx+c

5. -Step 3Draw the axis of symmetry x=2 -Step 4Identify 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 ofSquare 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. 72 = 1. 36 2 =    6 2 6 = = 2. 6 24 6 4 =   2 4

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

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

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 = 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.

21. 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

22. 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 Completing the square

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)