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: (x, y) = 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 = x = b 2a = y = f b 2a x = b 2a = = y = f b (x, y) = (x, y) = 2a =

3 Vertex form for a quadratic function (Let s think about absolute value functions again first to help us understand this.) Recall: Vertex form for the equation of an absolute value function is f (x) = a x h + k What do the constants in the equation represent? Graph the following absolute value equations using the vertex form. 1. f (x) = 2 x f (x) = 1 3 x +1 5 vertex = vertex = slope of the right branch of the graph = slope of the right branch of the graph =

4 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 = 1 2 (x + 3)2 2 (h,k) = (h,k) = (x, y) = (x, y) = a = a =

5 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 =

6 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

7 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

8 5.5 Completing the Square 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 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 : (h, k): (x, y): a =

9 2 2.) y = x + 6x 8 Vertex Form : (h, k): (x, y): a = 3.) y = x 2 + 4x 1 Vertex Form : (h, k): (x, y): a =

10 4.) y = 2x 2 12x + 14 Vertex Form : (h, k): (x, y): a = 5.) y = 4x 2 6x +1 Vertex Form : (h, k): (x, y): a =

11 6.) y = 3x 2 6x 8 Vertex Form : (h, k): (x, y): a =

12 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

13 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

14 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

15 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)

16 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

17 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.

18 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

19 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 =

20 Use the quadratic formula, factoring or taking square roots to solve for x in each equation. Use your graphing calculator to check your solutions. (Graph the equation using y =, then use 2 nd Calc, 2: Zero to find where the graph crosses the x-axis.) 1. 5(x 2) 2 +1 = x 2 + 5x 3 = x 2 + 5x +1 = 0

21 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

22 5.6 Motion Problems

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25 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?

26 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:

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

28 5.7 Solving Quadratic Inequalities You can easily solve quadratic inequalities by using the following algebraic method. Step 1: Step 2: Step 3: Step 4. Step 5: Factor the quadratic. Find the zeros of the quadratic and use them as boundaries for your solutions. Plot the boundaries on a number line. Use a test point to test the intervals between the boundaries. Summarize your solutions as a compound inequality or using set notation. FYI: What is set notation?? 3 x 5 can be written as [3,5] 1 x 10 can be written as [-1, 10] 4 < x < 7 can be written as (4, 7) 1 x < 8 can be written as [-1, 8) 5 < x 6 can be written as (5, 6] Use a bracket [ ] for less than or equal to, use a parenthesis ( ) for less than. Easy! 1. 2x 2 7x x 2 16x + 5 0

29 3. x 2 12x < x 2 + x + 5 < 0 (If you can t easily factor, use the quadratic formula to find the zeros.) J

30 5.8 Modeling with Quadratic Functions WRITING EQUATIONS OF QUADRATIC EQUATION 1.) Write an equation in vertex form for the parabola shown. Recall vertex form: y = a(x h) 2 + k 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. Recall standard form: ax 2 + bx + c = 0

31 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 intercept form of the equation of the parabola with x- intercepts of ( 1 3,0) and ( 1 5,0) and passes through the point (-1,2). Recall intercept form: y = a(x r 1 )(x r 2 )

32 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.

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