When Is Factoring Used?

Transcription

1 When Is Factoring Used? Name: DAY 9 Date: 1. Given the function, y = x 2 complete the table and graph. x y A ball is thrown vertically upward from the ground according to the graph below. The height is y and the time in seconds is x. a. Given the graph, complete the table. x y b. Does this graph represent a function? Why/Why not? c. The input is the and the output is the. d. What is the maximum height of the object? e. How could you find determine when the object hits the ground?

2 2. A ball is thrown vertically upward from the ground according to the function below (the height is y and the time in seconds is x). Function: y = 2x 2 + 4x + 16 a. Use the instructions below to determine when the object hits the ground. Questions Work 1 The point where the object hits the ground is the of the graph. 2 Write the function. Every x-intercept has a y-coordinate of. 3 So in order to determine the x-intercept, substitute in for y. 0 = 2 ( ) 4 Factor out the negative 2. Then factor the trinomial. 0 = 2 ( ) ( ) 5 Solve for x using the zero product property. 6 The object hits the ground after seconds.

3 Name Date: Block: Below are three quadratic functions and their graphs y = (x + 4)(x 2) Below you have three quadratic functions and the graph. What happens to the way the y- values change when we change a? 1. y = (x + 3)(x 3) 2. y = 2(x + 3)(x 3) 3.

4 EXAMPLE ONE Graph the following quadratics: 1. y = 2x 2 + 8x y = 1 2 x2 3x y =.25(x + 6) y = 3(x 2)(x + 1)

5 GRAPHING SUMMARY Method Formulas or Shortcuts Standard Form Vertex Form Intercept Form

6 Name Date: Block: Finding the vertex point: 1. Graph the function in Y1. Press Graph. 2. Make sure you can see the vertex point. If not, change the window. 3. Press 2 nd TRACE (CALC) 4. Choose 3: minimum or 4: maximum a. Choose minimum if the graph opens up b. Choose maximum if the graph opens down 5. Scroll to the left side of the vertex point. Press ENTER 6. Scroll to the right side of the vertex point. Press ENTER. 7. Press ENTER one more time Use your calculator to find the vertex point for each of the following quadratic functions. 1. y = 2x 2 5x y = -x 2 + 6x y = 4.8x x y =.05x 2 4x + 5 Finding the zeros: 1. Press 2 nd Trace 2. Choose number 2: zero 3. Scroll to the left of the zero. Press ENTER. 4. Scroll to the right of the zero. Press ENTER. 5. Guess where the zero is. Press ENTER. 6. Calculator will now display zero x = # y = # Use your calculator to find the zeros for each of the following quadratic functions. 5. y = 2x 2 5x y = -x 2 + 6x y = 4.8x x y =.05x 2 4x + 5

7 Name Date: Block: Evaluating at a given x - value: Press 2 nd Trace 1. Choose 1: value 2. Type the number you wish to plug in 3. Press Enter Use your calculator to evaluate the function for the given x values. 9. f(x) =.05x 2 4x + 5 a. f(-5.32) b. f(1/4) c. f(100) 10. h(x) = -x 2 + 6x + 10 a. h(-5.32) b. h(1/4) c. h(100) 11. g(x) = 4.8x x a. g(-5.32) b. g(1/4) c. g(100) Solving Equations for a given y value 1. In Y2 =, type the number you wish to substitute for y 2. Press Graph 3. Press 2 nd Trace 5: intersect 4. Scroll close to the intersection point 5. Press ENTER ENTER ENTER 6. The calculator will now display the Intersection x = # and y = # Use your calculator to find the zeros for each of the following quadratic functions. 12. y = 2x 2 5x + 7; y = y = -x 2 + 6x + 10; y = y = 4.8x x + 2.9; y = y =.05x 2 4x + 5; y = -5

8 1. y = 12(x + 2)(x 4) 2. y = 3(x 1)(x 2) Name Date: Block: 3. y = (x + 2)(x + 5) 4. y = 3(x 1)(x 3) 5. y = 14(x 2)(x + 8) 6. y = 12 ( x 3)(x + 4)

9 Name Date: Block: 1. y = 2(x 2) y = 3(x + 2) y = (x 4) y = 12 (x + 7) y = 14(x 1) y = 2(x 3) 6

10 Name Date: Block: 1. y = x 2 4x y = x 2 6x 8 3. y = 2x 2 + 4x 9 4. y = x 2 x y = x 2 3x y = 1/2 x 2 + 5x + 7

11 FACTORING REVIEW Name: Date: I. Binomial Difference Squared (A B) 2 Ex 3: Ex 4: Formula: II. Conjugates: (A + B) (A B) Same binomial except addition/subtraction signs Ex 5: Ex 6: Formula:

12 III. Perfect Square Trinomials (A = 1) Factoring Perfect Square Trinomials General Form of a Trinomial:, where A, B, and C are constants. Ex 7: Ex 8: The square root of a perfect square trinomial is a:. Perfect Square Trinomial Test #1 If A = 1 and, then the trinomial is a perfect square trinomial!! It will factor into a. Factor it by: IV. Perfect Square Trinomials (A = 1) If A = 1, different properties hold true. If A = 1 and, then the trinomial might be a perfect square trinomial!! It might factor into: You know that the and terms will work, but You must check to make sure that:.

13 Factoring Trinomials V. Trinomials of the Form: (not perfect square trinomials). Recall ( ) ( ), Ex 1: x 2 + 3x + 2 Ex 2: x 2 5x + 6 Method 1. Write down two pairs of parentheses. 2. Determine the factors of C. 3. Find the combination of factors that will add/subtract to equal B. 4. Place the values into the parentheses 5. Check using FOIL. Ex 3: x 2 2x 8 Ex 4: x 2 + 7x - 18

14 Factoring Trinomials: Ax 2 + Bx + C, A = 1 VI. Factoring Trinomials, where A is prime. Method 1. Find the factors of A. 2. Place them in the first term of both parentheses. 3. Find all factors of C. 4. Determine which combination of factors will yield the middle term when multiplied and added. 3. Check. Ex 1: 2x x + 5 Ex 2: 3x 2 4x 7 II. Factoring Trinomials, where A is not prime. Method 1. Find the factors of A and C. 2. Determine which combination of factors will yield the middle term when multiplied and added. 3. Check. Ex 3: 6x 2 19x + 15 Ex 4: 6x 2 2x 8

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