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Quadratic Journal

Quadratic Journal. By : Christian Aycinena. Factoring any Polynomial:. First of all, a polynomial is a mathematical expression in which consists of two or more terms. How to factor a Polynomial? First find the GCF(Greatest Common Factor) Divide all by the GCF Do Backward FOIL .

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Quadratic Journal

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  1. QuadraticJournal By: Christian Aycinena

  2. Factoring any Polynomial: • First of all, a polynomial is a mathematical expression in which consists of two or more terms. • How to factor a Polynomial? • First find the GCF(Greatest Common Factor) • Divide all by the GCF • Do Backward FOIL

  3. Examples: 1. 2x2-4x+6 GCF is 2 so we divide all by 2. 2(x2-2x+3) 3x3+6x2+9x –the GCF is 3x so we divide all by 3x. 3x(x2+2x+3) 2. 3. 2x6-18x5-36x4- the GCF is 2x4 so we divide all by 2x4. 2x4(x2-9x-18)

  4. Quadratic Functions Quadratic Function: A function that have a standard form of y = ax2+bx+c. “A” can never be 0.The graph of a quadratic function is a parabola.

  5. The difference between a quadratic and a linear function is... • 1. Degree—y =mx1+b y=ax2+bx+c • 2. Graph of each function is different in shape. *Linear functions has a linear • *Quadratic function has a parabolic shape • 3. Quadratic standard form: y=ax2+bx+c Linear: y=mx+b

  6. Examples: Are this functions linear or quadratic? A. y= 5x + 7 E. B. y= 7x2+8x+33 C. y= 2x2+10x D.

  7. Answers! A.Is linear B. Is quadratic C. Is quadratic D. Is Linear E. Is Quadratic

  8. How to graph a quadratic function • We know the standard form which is y =ax2+bx+c but the graphing form is a(x+b)2+c =0 • A- It changes the steepness of the parabola. • If a < 0- Goes down. • If a >0 Goes up. • If a <1 it is wider If a >1 it is steeper

  9. B- Moves it right or left, b units, moves it opposite signs(positive-left, negative-right) C- Moves vertex up or down, c units, (positive-up, negative-down)

  10. Examples: 1.y=-2(x-4) 2+5 This in graph will be steep, it will face down, it will be 4 to the right and it will be 5 up. Not real scale

  11. 2.y=0.5(x+3) 2-1 This graph will be wide, and will face up. The vertex will be at 3 to the left and it will be down 1. Not real scale

  12. 3.y= (x) 2-6 This graph will be no so wide as last but not so steep as the first one. It will face up. It will be in the middle and 6 down. Not real scale

  13. Maximum and Minimum values When a parabola opens down, where the vertex is it has a maximum value. In the other hand, When a parabola opens, like a u, where the vertex is it has a minimum value. Minimum value

  14. How to solve a quadratic equation by graphing it • 1. Set ax2+bx+c=0 • 2. Graph the functions. – • Make a t-table • Find the Vertex using this formula x= -b/2a • Pick 2 points to the left of the vertex and two to the right. • Solve it and Graph it. • Find the x-values where it crosses the x-axis(this is your solution). • Solution:are the x-values where the parabola crosses the x-axis. If the parabola doesn´t cross the x-axis there is no soultion.

  15. Square Roots • Get x2 by itself. • Make sure there is no x by itself • Find the square root of both sides and Don´t forget the two solutions of a square root(+ or -)

  16. 1. Examples: X2 = 81 X2 = 81 X=+9 or -9 3. 16X2 -3= 61 +3 +3 16X2 = 64 16 X2 = 4 X= +2 or -2 2. X2 -7= 9 +7 +7 X2 = 16 X2 = 16 X=+4 or -4

  17. Quadratic Equations using Factoring! a b c • First find the GCF(Greatest Common Factor) • Divide all by the GCF • Do Backward FOIL • A) Multiply a – c • B) Find two numbers that add up to b and also that multiply to the product you got. • C) Put the numbers you got over a • D) Reduce • Open parenthesis, write what you got when you reduce and put it equal to O. Finally write down your solutions.

  18. Examples: 1.

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  21. Solving a quadratic equation using Completing the square • Get x2= 1 • Get C by itself. • Find b, divide by 2 and square it. • Add (b/2)2 to both sides; • Factor (x+b/2)2 in one side. • Square root both sides. • Use both roots(+,-) and solve for x

  22. Examples:  x2-10x+16=0 -16 -16 so x2-10x=-16 then find b, divide b/2, then square (b/2) = (-10/2)2 =(-5)2= 25 +25+25 so then you factor it and get (x-5)2= 9 after you get the square root of both it will be x-5= +3 or -3 +5 +5 +5 x=8,2 1. Now guide your self! a2+2a-3= 0 +3 +3 a2 +2a= 3 a2+2a+1=3+1 (a+1)2=4 SQR a+1= +2, -2 -1 -1 -1 a=1, -3 3. 2. x2-8x+12=0 , subract 12 from both sides x2-8x =-12 x2-8x+16=-12+16 (x-4)2=4 SQR x-4 = +2, -2 X= -6, -2

  23. Quadratic Formula • Determine the values of a, b and c. • Substitute the values into the quadratic formula • Find the two solutions. • Before solving use the rule of the discriminant that says if the value of the discirnminat is positive the equation will have two solutions, but if it is zero the equation will have only one solution and finally if the value of the discriminant is a negative number than the equation has no real solutions.

  24. Examples: 1.

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