Solve the system of linear equations. Warm-up 5/8/08 Solve the system of linear equations. 4x + 3y = 12 2x – 5y = -20 (0,4) B(6,0) (2,10) 1

Up Next Chapter on Quadratic Equations… Getting ready for Algebra II Last test will be over this chapter. It will count as a regular test. For some of you, it will be your last chance to pull up your grade.

Unit 7 Topic: (Chapter 7) Quadratic Equations and Functions Key Learning(s): How to set up and solve quadratic problems Unit Essential Question (UEQ): How do you solve quadratic equations and functions?

Concept I: Modeling Data with Quadratic Functions Lesson Essential Question (LEQ): How do you graph and apply quadratic functions? How do you locate a quadratic function that is shifted? Vocabulary: Parabola, quadratic functions, standard form, Axis of symmetry, vertex, maximum value

Concept II: Square Roots Lesson Essential Question (LEQ): How do you use square roots when solving quadratics? Vocabulary: Square root, principal, square root, negative square root, perfect squares

Concept III: Solving Quadratic Equations Lesson Essential Question (LEQ): How do you determine whether a quadratic equation has two solutions, one solution, or no solutions? Vocabulary:

Concept IV: Using the Quadratic Formula Lesson Essential Question (LEQ): When would you use the quadratic formula to solve a quadratic equation? Vocabulary: Quadratic formula, vertical motion formula

Concept V: Using the Discriminant Lesson Essential Question (LEQ): How do you use the discriminant to find the number of solutions of a quadratic equation? Vocabulary: Discriminant

Introduction On a piece of paper (in your notes) -Show the path of an arrow if it is aimed horizontally. -How does the path change if the arrow is aimed upward? -Name its shape WARM-UP COMPLETE TEST COPY SLM INTRODUCTION 4.1 NOTES/EXAMPLES 4.1 WORKSHEET WORK ON PROJECTS/CHECK PROGRESS

7.1: Modeling Data with Quadratic Functions LEQ: How can you tell before simplifying whether a function is linear, quadratic, or absolute value? Remember HOW? y = (x – 3)(x + 2) f(x) = x(x+3) (x+4)(x-7) (2b-1)(b-1) http://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/Operations-with-Polynomials-and-FOIL.lesson x2 – x – 6 X2 + 3x X2 – 3x – 28 2b2 – 3b + 1

Quadratic Function A function that can be written in the form y = ax2 + bx + c, where a≠0. The graph of a quadratic function looks like a “u” (or part of one). The graph of a quadratic function is called a parabola.

Linear or Quadratic? A function is linear if: the greatest exponent of a variable is one A function is quadratic if: the greatest exponent of a variable is two

Linear or Quadratic? y = (x – 3)(x + 2) f(x) = x(x + 3) f(x) = (x2 + 5x) –x2 y = (x – 5)2 y = 3(x – 1)2 + 4 h(x) = (3x)(2x) f(x) = ½ (4x + 10) y = 2x – (3x – 5) f(x) = -x(x – 4) + x2 y = -7x Quadratic Linear

Ex1) Make a table of values and graph the quadratic functions y = 2x2 and y = -2x2. What is the axis of symmetry for each graph? What effect does a negative sign have on the shape of a quadratic function’s graph? Graph y = -1/3 x2, y = ½ x2, y = -x2 Compare the width of the graphs above. How could you quickly sketch the graph of a quadratic equation?

Properties of parabolas Important parts: Vertex – The point at which the function has a maximum or minimum Axis of Symmetry – Divides a parabola into two parts that are mirror images Talk about things in real-life that use parabolas (bridges)

Maximum or Minimum If a parabola opens down, does it have a maximum or minimum for the vertex? If a parabola opens up, does it have a maximum or a minimum for the vertex?

Practice Section 7.1 p. 321-322 # 2 – 40 Even Assignment p. 321 – 322 #1 – 11 odd, 17, 19, 25, 29, 31, 35 - 39

Warm-up 2/19/08 Re-write each equation without parenthesis. Y = (x + 1)2 + 4 Y = 2(x – 5)2 Y = -(x – 3)2 + 6 Y = -3(x – 7)2 Y = 7(x + 4)2 Y =x2 + 2x + 5 Y = 2x2 – 20x + 50 Y = -x2 + 6x – 3 Y = -3x2 + 42x – 147 Y = 7x2 + 56x + 112

Reminders Late projects? Late word problems?

Assignment Long weekend homework? Section 5.2 p. 208-209 #1-31 odd

Graphic organizer (vertex form) Refresh Graphic organizer (vertex form)

Warm-up 2/20/08 Write the equation of the parabola shown (in vertex and standard form):

Write the equation of the parabola shown:

p. 210 #38 – 40, 42 Worksheet Practice Homework p. 210 #38 – 40, 42 Worksheet Practice

§5.3: Vertex vs. Standard Form LEQ1: How do you find the vertex form of a function written in standard form? LEQ2: How do you transfer functions from vertex to standard form? Which is better, standard or vertex form? Standard Form Vertex Form Y = -3x2 – 12x – 8 y = -3(x + 2)2 + 4

When the equation of a function is written in standard form, 1) the x-coordinate of the vertex is –b/2a. 2) To find the y-coordinate, you substitute the value of the x-coordinate for x in the equation and simplify.

Ex1. Write the function y = 2x2 + 10x + 7 in vertex form. x-coordinate: -b/2a -10/2(2) = -10/4 =-5/2 or - 2.5 y-coordinate: 2(-2.5)2 + 10(-2.5) + 7 -5.5 Substitute the vertex point (-2.5,-5.5) into the vertex form y = a(x – h)2 + k a from above y = 2(x + 2.5)2 – 5.5

Ex2. Write y = -3x2 +12x + 5 in vertex form. x-intercept: -b/2a = -12/2(-3) = -12/-6 = 2 y-intercept: y = -3(2)2 + 12(2) + 5 = 17 Re-write in vertex form: y = -3(x – 2)2 + 17

Write in vertex form:

Write in vertex form:

Write in vertex form:

Write in vertex form:

From vertex to standard form: To change an equation from vertex to standard form, you have to multiply out the function. y = 3(x -1)2 + 12 y = 3(x – 1)(x – 1) + 12 y = 3(x2 – 2x + 1) + 12 y = 3x2 – 6x + 3 + 12 y = 3x2 – 6x + 15

Graphic Organizer

Assignment Section 5.3 p.212-213 #1-4 all, 8-18 even, 22-30 even, 36

Warm-up 5.3 The Smithsonian Institution has a traveling exhibit of popular items. The exhibit requires 3 million cubic feet of space, including 100,000 square feet of floor space. How tall must the ceilings be? The exhibit also requires a constant temperature of 70ºF, plus or minus 3º. write an inequality to model this temperature, T. 30 feet 67 less than or equal to T less than or equal to 73

Warm-up 5.4 & 5.5 Find the inverse of each equation. y = -√(x + 2)

Warm-up 5.5 Multiply the two binomials: (y – 7 )(y + 3) (2x + 4)(x + 9) (4b – 3)(3b – 4) (6g + 2) (g – 9) (7k – 5)(-4k – 3)

§5.5: Factoring Quadratic Equations LEQ: When is the quadratic formula a good method for solving an equation? Which is better, standard or vertex form? One way to solve a quadratic equation is to factor and use the Zero-Product Property. *For all real numbers a and b, if ab = 0 then a = 0 or b = 0.

To solve by factoring, first write an equation in standard form To solve by factoring, first write an equation in standard form. Factoring x2 + 5x + 6 requires you find two binomials of the form (x + m)(x + n), whose product is x2 + 5x + 6. M and N must have a sum of 5 and a product of 6.

Steps: List the factor pairs whose product is 6 Find two of those factors whose sum is 5 Ex. Factor pairs of 6: 6 x 1 2 x 3 -1 x -6 -2 x -3 Only one pair has a sum of 5: 2 and 3 Thus, m & n are 2 & 3… (x + 2)(x + 3) *Always check by using the FOIL method!

Ex. Factor x2 – 7x + 12 Find factor pairs of 12 1x12 2x6 3x4 -1x-12 -2x-6 -3x-4 Which factor pair has a sum of -7? -3x-4 So, put -3 and -4 in where the m & n would be. (x – 3)(x – 4) Check with FOIL.

Factor the trinomials. x2 + 12x + 20 x2 – 9x + 20 (x + 10)(x + 2)

Warm-up 5.5cont. Find the inverse of each function. ±√(x/2 – 1) Y = -1/2 x Y = 1/3x2 - 2 Y = 2x2 + 2 Y = -2x Y = √(3x + 6)

§5.5 Continued Solve each equation by factoring. x2 + 6x + 8 = 0 x2 – 2x = 3 2x2 + 6x = -4 X = -4,-2 X = 3, -1 X = -1,-2

Solving by finding roots: Quadratic equations can also be solved by finding the square roots. *This method is effective when there’s no “b”. Ex. 0 = -16x2 + 1600 -1600=-16x2 16 16 100 = x2 x = ±10 Determine the reasonableness of a negative answer based on the situation.

Word Problem A smoke jumper jumps from a plane that is 1700 ft above the ground. The function y = -16x2 + 1700 gives a jumpers height y in feet after x seconds. How long is the jumper in free fall if the parachute opens at 1000 ft? How long is the jumper in free fall if the parachute opens at 940 ft? 6.6 seconds 6.89 seconds

Roots using the calculator Roots, also called “zeros” are really the points where a quadratic equation intercepts the x-axis (where x = 0). To find the zeros using a calculator: 1) enter the quadratic function under y = 2) 2nd calc … zeros… 3) left bound? Right bound? Enter…

Find the roots of each equation by graphing. Round answers to tenths x2 – 7x = -12 6x2 = -19x – 15 5x2 – 7x – 3 = 8 1 = 4x2 + 3x X = 3,4 X = -1.5, -1,7 X = -0.9, 2.3 X = -1, 0.3

What if you can’t factor it? If you’re having trouble factoring a quadratic equation, you can always use the quadratic formula. (Graphic Organizer)

Solve using any method. 5x2 = 80 X2 – 11x + 24 = 0 12x2 – 154 = 0

T.O.T.D. Answer the LEQ’s. 1) When is the quadratic formula a good method for solving an equation? 2) Which form is better? Standard or vertex form? Are some situations easier to use one or the other? Explain. (this question comes from several days)

Warm-up 5.6 Solve each equation. Give an exact answer if possible. Otherwise write the answer to two decimal places. x2 – 4x – 21 = 0 2x2 – 3x = 0 (x – 3)(x + 4) = 12 (x + 1)(x – 2)(2x + 1) = 0 -3,7 0, 1.5 -5.42, 4.42 -1, 2, -1/2

What if you can’t factor it? If you’re having trouble factoring a quadratic equation, you can always use the quadratic formula. (Graphic Organizer)

§5.6: Complex Numbers LEQ: How are complex numbers used in solving quadratic equations? What do you know about the graph? From previously, what if you had the equation: x2 + 25 = 0 You end up taking the √ of a negative number! (Calculator won’t work)

The Imaginary Number In order to deal with the negative square root, the imaginary number was “invented”. Imaginary Number: i defined as √-1 For now, you’ll probably only use imaginary numbers in the context of solving quadratics for their zeros. From the web…

i Imaginary Number i is the symbol for the imaginary number. It is a complex number whose square root is negative or zero. Rene Descartes was coined the term in 1637 in his book La Giometrie. The numbers are called imaginary because they are not always applied in the real world. i

Imaginary Number Applications In electrical engineering, when looking at AC circuitry, the values of electrical voltage are expressed as complex imaginary numbers known as phasors. Imaginary numbers are used in areas such as signal processing, control theory, electromagnetism, quantum mechanics and cartography.

Imaginary Number In mathematics Imaginary Numbers,also called an Imaginary Unit, can be found when working with quadratic functions. An equation like x2+1=0 has an imaginary root, and requires the use of the quadratic formula to solve it.

The Discriminant Whether or not you end up with a complex number as an answer depends solely on the discriminant. The discriminant refers to the part of the quadratic equation that is under the square root.

Nature of the solutions I. If the discriminant is positive -There are two real solutions -The graph of the equation crosses the x-axis twice (has two zeros) If the discriminant is zero -There is one real solution -the graph of the equation only touches the x-axis once (has one zero)

If the discriminant is negative -There is no real solution -There are two imaginary solutions -The graph never touches the x-axis. Example 1 y = x² + 2x + 1 a = 1 b = 2 c = 1 Discriminant: 2² - 4 • 1 • 1 = 4 − 4 = 0 Since the discriminant is zero, there should be 1 real solution to this equation. Also, the graph only touches the x-axis once.

= x² − 2x + 1 y = x² − x − 2 y = x² − 1 y = x² + 4x − 5 Calculate the discriminant to determine the number and nature of the solutions of the following quadratic equation: = x² − 2x + 1 y = x² − x − 2 y = x² − 1 y = x² + 4x − 5 y = x² + 4x + 5 y = x² + 4 y = x² + 25

Simplifying complex numbers: -1 -i 1 i

Complex Numbers A number of the form a + b(i) , where a and b are real numbers, is called a complex number. Here are some examples: 2 – i, 2 – √3i The number a is called the real part of a+bi, the number b is called the imaginary part of a+bi.

Operations with Complex Numbers Adding & Subtracting them: Just like combining like terms Ex. 3i + -1i = 2i (5 + 7i) + (-2 + 6i) Combine like terms, simplify 5 – 2 + 7i + 6i 3 + 13i

Multiplying Distribute, combine like terms, simplify Ex) (5 + 7i)(-2 + 6i) 10 + 30i – 14i + 42i2 10 + 16i + 42(-1) 10 + 16i – 42 -32 + 16i

Warm-up 5.7 Simplify each expression 1) √-25 (2 + 3i)(3 – 4i) 1) √-25 (2 + 3i)(3 – 4i) Multiply. (x + 1)(x + 1) 5) (x + 6)(x + 6) (x – 3)(x – 3) 6) (2x – 1)(2x – 1) 5i 18 + i x2 + 2x + 1 x2 – 6x + 9 x2 + 12x + 36 4x2 – 4x + 1

§5.7: Completing the square LEQ: How is completing the square useful when solving quadratic equations? Binomial Squared (x + 5)2 = x2 + 2(5)x + 52 = x2 + 10x + 25 (x – 4)2 = x2 + 2(-4)x +(-4)2 = x2 – 8x + 16 (x + b/2)2 = x2 + 2(b/2)x + (b/2)2 = x2 + bx + (b/2)2

*This method is useful for making vertex form. Find the b-term. The process of finding the last term of a perfect square trinomial is called completing the square. *This method is useful for making vertex form. Find the b-term. Divide the b-term by 2 The square of this will be c. Try these: 1) x2 + 2x + ___ 2) x2 – 12x + ___ 1 36

Ex1. Solve by completing the square. x2 = 8x – 36 Write the equation with all x-terms on one side: x2 – 8x = - 36 Complete the square (add to both sides): x2 – 8x + (-4) 2 = -36 + (-4)2 Re-write: (x – 4)2 = -36 + 16 (x – 4)2 = -20 It would be easy to graph this in vertex form.

To solve the equation, continue algebraically. (x – 4)2 = -20 √(x – 4)2 =√-20 (x – 4) = ± √-20 x = 4 ± √-20 x = 4 ± √(2)(2)(5)I x = 4 ± 2i√5 The two solutions are: x = 4 + 2i√5 x = 4 – 2i√5

Assignment Quiz 5.7

Warm-up Test Classify 3x(2x) as linear, constant, or quadratic. Re-write the equation of the parabola in vertex form: y = x2 + 8x + 12. Find the absolute value of 6 – 9i. Simplify -2(2 – 4i) – 8(5 + 2i). Find the coordinates of the vertex for the graph of y = x2 + 10x + 2

Warm-up Test (2) How do you use the vertical line test to determine if a graph represents a function? Find f(g(2)) if f(x) = 2x + 2 and g(x) = 9x. Graph the inequality y > 2x – 1. How do you solve a system of three equations in three variables? http://apps.collegeboard.com/qotd/question.do