Algebra II Chapter 5

Transcription

1 Algebra II Chapter 5

2 5.1 Quadratic Functions The graph of a quadratic function is a parabola, as shown at rig. Standard Form: f ( x) = ax2 + bx + c vertex: b 2a, f b 2a a < 0 graph opens down a > 0 graph opens up axis of symmetry: b x = 2a Graph each function given in Standard Form: Larson Text, Ch 5, p y = x 2 + 4x y = x 2 + 6x +11 a = a = b = b = b 2a = f b 2a b 2a = = f b (x, y) = (x, y) = 2a =

3 Vertex Form: f ( x ) = a ( x h ) 2 + k vertex: (h, k) axis of symmetry: x = h a < 0 graph opens down a > 0 graph opens up Graph each function given in Vertex Form. 1. y = (x 1) y = (x + 3) 2 2 (h,k) = (h,k) = (x, y) = (x, y) = a = a =

4 Intercept Form: f (x) = a(x r 1 )(x r 2 ) x-intercepts: (r 1,0) and (r 2,0) vertex: a < 0 a > 0 r 1 + r 2 2, f r 1 + r 2 2 graph opens down graph opens up Graph each function in Intercept Form. 1. f (x) = (x 2)(x + 4) 2. f (x) = 2(x +1)(x 5) r 1 = r 1 = r 2 = r 2 = vertex = vertex = a = a =

5 5.2 Solving Quadratic Equations by Factoring Zero Product Property Let A and B be real numbers or algebraic expressions. If AB = 0, then A = 0 or B = 0. Solve each equation by using the Zero Product Property: 1. 0 = (x 1)(x 3) 2. 0 = x(x + 4) 3. 0 = x = x 2 + 3x x 2 17x + 45 = 3x x 2 +12x 7 = = 6x 2 16x x 2 5 = 0

6 5.3 Solving Quadratic Equations by Finding Square Roots Simplify expressions using properties of square roots: 1) 48 2) 90 3) ) ) 6 i 10 6) Solve the quadratic equation by finding square roots. 7) 2x 2 +1 = 17 8) x 2 9 = 16 9) 4x = 23 10) 5(x 1) 2 = 50 11) 1 2 (x + 8)2 = 14

7 5.5 Completing the Square Warm Up: 2 1.) f ( x) = x 4x 12 Vertex: x intercepts: 2.) y = ( x 2) 2 16 Vertex: X intercepts: Standard Form: 3.) Factor the Perfect Square Trinomials A.) x 8x + 16 = B.) x + 5x + = C.) x 7x + = 4 Can you see any pattern on how the second term in factored form is related to the middle term of the original quadratic? Find the value of c that makes the quadratic equation a perfect square trinomial. Then write the quadratic in vertex form. 2 1.) y = x 14x + c ) y = x + x + c 3

8 Completing the Square to Graph a Quadratic Function Rewrite the equation in vertex form by completing the square. Find the vertex. Then solve for the x intercepts. Verify your x intercepts on the calculator. Graph the parabola. 2 1.) y = x + 10x 3 Vertex Form : Vertex: X intercepts: 2 2.) y = x + 6x 8 Vertex Form: Vertex: X intercepts :

9 3.) y = x 2 + 4x 1 Vertex Form: Vertex: X intercepts : 4.) y = 2x 2 12x + 14 Vertex Form: Vertex: X intercepts :

10 5.) y = 4x 2 6x +1 Vertex Form : Vertex: X intercepts : 6.) y = 3x 2 6x 8 Vertex Form : Vertex: X intercepts :

11 Completing the Square to solve a Quadratic Equation Solve the following equations by completing the square. 1.) x 2 12x = 28 2.) x 2 + 3x 1 = 0 3.) 3x x = 27 4.) 4x 2 40x 8 = 0 5.) 3x 2 26x + 2 = 5x ) 2x 2 + 3x +1 = 0 7.) 4x 2 2x = 5 8.) 3x 2 + 5x = 7

12 5.4 Complex Numbers (part 1) What happens when you try to solve: x 2 = 1??? By definition: i = 1 i 2 = ( 1) 2 = 1 Simplify the following radicals. Give your answer in terms of i From Larson Textbook, page 272

13 Solving Equations over the Complex Numbers Solve the equation for x, giving your answer in terms of i. 1.) x = 0 2.) 2x = 10 Check: Check: 3.) 3x 2 +10x = 26 4.) 6x 2 2x + 2 = 4x 2 + x 5.) 1 2 (x +1)2 = 5 6.) 6(x + 5) 2 = 120

14 Plot the complex numbers in the complex plane: imaginary axis i i 3. 2i 4. -i + 7 real axis Adding and subtracting complex numbers: Combine the real parts and combine the imaginary parts. 1. (3+ 4i) + (6 + i) 2. (1 i) (7 + 3i) 3. (6 2i) (5 + i) (10 + 5i)

15 5.4 Complex Numbers (Part 2) Multiplying Complex Numbers Simplify each expression as a complex number in standard form: a + bi 1. 3i(5 i) 2. 7i(3 2i) 3. (2 + 3i)(5 6i) 4. (2 5i)(2 + 5i) 5. (1+ i)(1 i) Write each expression as a complex number in standard form: a + bi i i 1 2i i 3 i 4. 8 i 8 + i i 2i 6. 6 i

16 Why do we need complex numbers? Complex numbers are at the heart of understanding Fractal Geometry. See pages 275 and 276 of your textbook. Fractal Geometry is used to model a variety of natural phenomena. Check out this video: Complex numbers are also used in electronics to describe electrical circuits. Complex numbers are used in a variety of higher level mathematics.

17 5.6 The Quadratic Formula Find the x-intercepts for the following equations by completing the square : y = x 2x + 7 y = ax + bx + c

18 Quadratic Formula: y = ax 2 + bx + c x = b ± b2 4ac 2a Use the quadratic formula to solve for x in each equation. 1. x 2 + 3x 2 = x 2 + 2x + 9 = 0 a = b = c = a = b = c = x = x = 3. 5x 2 + 9x = x 2 + 5x +1 a = b = c = x =

19 Use the quadratic formula, factoring or taking square roots to solve for x in each equation. Use your graphing calculator to check your solutions. 1. 5(x 2) 2 +1 = x 2 + 5x 3 = x 2 + 5x +1 = 0

20 5.6 The Discriminant For each equation, find the value of the discriminant, determine how many solutions, and then find the solutions. Check your answers using your graphing calculator. 1. x 2 6x +10 = 0 2. x 2 6x + 9 = 0 3. x 2 6x + 8 = 0

21 5.6 Motion Problems

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24 5.7 Graphing Quadratic Inequalities Warm up: Graph the following parabola by finding the vertex and the x-intercepts: 2 1.) y = x 2x 3 Vertex: x-intercepts: Now what if the original equation was written in the form: 2 y x 2x 3 How would that change the graph? Choose 2 points inside the parabola: Choose 2 points outside the parabola: 1.) 1.) Solution? Solution? 2.) 2.) Solution? Solution?

25 Graph the following quadratic inequality or system of quadratic inequalities: 1.) y < 2x 2 5x 3 Vertex: X intercepts: 2.) A.) B.) y x 2 y < x 4 2 x + 2 Equation A Vertex: x ints: Equation B Vertex: x ints:

26 A.) 3.) B.) y x y > x x 6 Equation A Vertex: x ints: Equation B Vertex: x ints:

27 5.7 Solving Quadratic Inequalities Solve the quadratic inequality by a graphing method. 1. 2x 2 7x x 2 + x + 5 < 0

28 Solve the quadratic inequality by an algebraic method. 3. 3x 2 16x x 2 12x < 32

29 5.8 Modeling with Quadratic Functions WRITING EQUATIONS OF QUADRATIC EQUATION 1.) Write an equation in vertex form for the parabola shown. 2.) Write the equation of the parabola in vertex form with vertex (-2,3) and passes through the point (2,-5). 3.) Write an equation in standard form for the parabola shown. 4.) Write the standard form of the equation of the parabola with x- intercepts of (-3,0) and (2,0) and passes through the point (-1,4). 5.) Write the standard form of the equation of the parabola with x- intercepts of ( 1 3,0) and ( 1,0) and passes through the point (-1,2). 5

30 6.) A study compared the speed x (in miles per hour) and the average fuel economy y ( in miles per gallon) for cars. The results are shown in the table. Speed,x Fuel economy,y a.) Graph a scatter plot of Fuel Economy vs. Speed on your graphing calculator. Do you think that a linear or quadratic regression model fits the data better? b.) Find a linear regression model for the data using your graphing calculator. Report the equation of the linear model below. Report the r 2 value for this model. c.) Find a quadratic regression model for the data using your graphing calculator. Report the equation of the quadratic model below. Report the r 2 value for this model. d.) Using the model that fits the data best, predict the fuel economy for a car travelling at a speed of 80 mph. e.) Find the speed that maximizes fuel economy.