16x y 8x. 16x 81. U n i t 3 P t 1 H o n o r s P a g e 1. Math 3 Unit 3 Day 1 - Factoring Review. I. Greatest Common Factor GCF.

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1 P a g e 1 Math 3 Unit 3 Day 1 - Factoring Review I. Greatest Common Factor GCF Eamples: A. 3 6 B C. 16 y 8 II. Difference of Two Squares Draw ( - ) ( + ) Square Root 1 st and Last Term Eamples: A B. 3 7 C III. Grouping Group 1 st two terms and last two terms with parentheses Take out a GCF from first and last grouping (Hint: your parentheses should be the same) Take out a GCF (the common parentheses) from that step A B C. a b ay by IV. Trinomials with Leading Coefficient of 1 E. 7 1 F G H. 4

2 P a g e V. Trinomials with Leading Coefficient Greater than A B. 5 3 C. 6 1 D VI. Sum or Difference of Two Cubes Patterns: a 3 + b 3 = (a + b)(a ab + b ) a 3 b 3 = (a b)(a + ab + b ) Steps: Draw - ( ) ( ) Signs - Same, Opposite, Always Positive To Get Little ( ) - Cube root each term To Get Big ( ) - Square 1 st, Multiply Middle, Square Last Factor the following a b a 64b 3 3

3 P a g e 3 Math 3 Unit 3 Day - Solving Quadratic Equations by Factoring and Imaginary Numbers Part 1 Quadratic Equations The graph is a parabola, a u-shaped figure. Parts of a Parabola: Verte Line of Symmetry/Ais of Symmetry Direction of opening X and Y intercepts Solutions: where the graph touches/crosses the. *Terms used for solutions of quadratic equations:,,. *There are three possible outcomes when solving quadratic functions: The Fundamental Theorem of Algebra: The degree of a polynomial (the highest eponent) indicates I. Solving Quadratic Equations Steps: Set the equation equal to zero (move everything to one side).. Factor the polynomial. Set each factor equal to zero and solve. *The number of solutions. Eamples:

4 P a g e II. Write the quadratic equation given the roots. -5, 6. 1, 3, , 5 III. Comple Numbers Eample: What is the square root of -5?

5 P a g e 5 If i 1, what is i? Which way is correct? i 1 i 1 Rules:. Eamples: (3 i) 4. ( i 5) 5. ( ) 6. i i i i i

6 P a g e 6 Math 3 Unit 3 Day 3 Imaginary Numbers Part, Completing the Square and Quadratic Formula I. Operations with Imaginary Numbers 6 i(3i 8). 3 4 i(5 8 i) (5i 9). (1 7 i)(10 9 i) 4.. ( i ) II. Equations III. Find the value of each variable. m 3 ( n 4) i 3 i. (3m 4) (3 n) i 16 3i ( m n) i (7m n) 9i 9 4. (10m 9 n) (5m 4 n) i 15 10i

7 P a g e 7 IV. Completing the Square A. Write the trinomial as the square of a binomial B. Find the value of C that makes the trinomial a perfect square. C 6 " ". C 8 " " C 4 " " C. Square Root Property ( 5) 7. 1 ( 3) V. Completing the Square A. Steps: Isolate the constant (move the constant to the other side).. Make sure the leading coefficient is 1 (if not divide through). Complete the Square (Divide the middle number by and square it). 4. Add that number to both sides of the equation. 5. Factor the left side and combine the right side (Short cut for factoring - Square root the first term, square root last term, and take the first sign). 6. Square root both sides of the equation. (Remember to put a ± on the right.) 7. Solve the equation. YOU SHOULD HAVE TWO ANSWERS FOR EACH PROBLEM.

8 P a g e 8 Eamples: VI. Quadratic Formula b b 4ac a Put the equation in standard form to find a, b, and c. Eamples:

9 P a g e 9 Math 3 Unit 3 Day 4 - Graphing Parabolas and Circles I. Parabolas - Up/Down y a( h) k a = Positive opens up a = negative opens down Verte (h,k) Eamples: Decide if the parabola opens up or down; state the verte, and graph. y ( 1) 4 Over 1 3 Up or Down Steps if Equation for Parabola is not in Standard Form (Similar to Completing the Square) Move the constant to the other side.. Make sure the leading coefficient is If not, divide through. Complete the square. Remember to balance the equation and add that number to both sides. 4. Factor the trinomial you created and combine like terms on the other side. 5. Solve for y. Decide the direction the parabola opens; write the equation in standard form, state the verte, and graph. y 4 6. Opens y 8 6 Opens Verte Verte

10 P a g e 10 y y Opens Opens Verte Verte II. Circles Write the equation of each circle in standard form and graph. Steps: Group all the like terms together.. Create two perfect square trinomials by completing the square AND balance the equation. Factor both trinomials. You now have the center and the radius y 18y + 64 = 0

11 y 59 = 0 + y 10 + y 5 = 0 U n i t 3 P t 1 H o n o r s P a g e 11

12 P a g e 1 Math 3 Unit 3 Day 5 Parabolas..graphing in a new way with Focus and Directri Understanding the Pieces Graph the point (3,1). Go eactly units up from the verte and place a point. Go eactly units down from the verte and place a point then draw a horizontal line through that point. Practice 1: Focus is (5,-3) Directri is y = 3 What is the verte? Which way does it open?

13 P a g e 13 Practice : Directri is y = Verte is (-1,1) What is the focus? Which way does it open? Practice 3: Verte is (-,-4) Focus is (-,-) What is the directri? Which way does it open? Practice 4: Verte is (4,6) Focus is (4,8) The length of the latus rectum is 8. Graoh the parabola. Practice 5: Verte is (10,-6) Focus is (10,-) The length of the latus rectum is 6. Graph the parabola. Graph the parabola and identify the verte, the focus, the directri, and the length of the latus rectum. 4y 13 Verte Focus Directri LR

14 P a g e 14. Graph the parabola and identify the verte, the focus, the directri, and the length of the latus rectum. 16y 4 0 Verte Focus Directri LR Graph the parabola and identify the following parts y Opens A.O.S. Focus Verte Value of a Directri

15 P a g e 15 Writing the Equation of a Parabola given the Focus and the Directri Focus (3, 8) Directri is y = 4. Focus is (7, -) Directri is y= 4 Write the equation of a Parabola given the Verte and the Focus. Verte is (-1,) Focus is (-1,0). Verte is (5,) Focus is (3,)

16 P a g e 16 Math 3 Unit 3 Day 6 - Optimization Steps: Set up two equations (do not use and y).. Solve the equation without the work ma/min in it for any variable. Substitute that epression into the other equation. 4. Know what your variables represent. 5. Enter into the calculator for y= and find the maimum or minimum. 6. Answer the question. Eamples: Find two positive numbers whose sum is 36 and whose product is a maimum.. Find two numbers whose difference is 8 and whose product is a minimum. A rectangle has a perimeter of 40 meters. Find the dimensions of the rectangle with the maimum area.

17 P a g e Loni has 48 feet of fencing to make a rectangular dog pen. If a house is used for one side of the pen, what would be the length and width for maimum area? 5. The circulation of the Charlotte Observer is 50,000. Due to increased production costs, the council must increase the current price of 50 cents a copy. According to a recent survey, the circulation of the newspaper will decrease 5000 for each 10 cent increase in price. What price per copy will maimize the income from the newspaper? 6. A baseball stadium normally can sell 10,000 hotdogs at a game if they sell them for $4.00 each. The also notice that if they raise the price $0.5 they will sell 500 fewer hotdogs. Determine the price they should charge to maimize revenue. 7. Marsha is making a bo to collect toys for the school toy drive. She cuts a 5 centimeter square from each corner of a rectangular piece of cardboard and folds the sides up to make a bo. If the perimeter of the bottom of the bo must be 50 centimeters, what should the length, width, and height of the bo be for a maimum volume?