Warm-up 4.1 1) At which vertex is the objective function C = 3x + y maximized? A(0,0); B(6,0); C(2,6); D(3,5); E(0,5) 2) Which point is a solution of y > x2 + 5? (2,10); (1,6); (0,4); (-1,4); (-3,0) 3) When f(x) x2 and g(x) = -2x 5, what is the value of f(g(-3))? Modeling Data with Quadratic Functions
How can you tell before simplifying whether a function is linear, quadratic, or absolute value? Remember HOW? 1) y = (x 3)(x + 2) 2) f(x) = x(x+3) 3) (x+4)(x-7) 4) (2b-1)(b-1) http://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/Operations-with-Pol ynomials-and-FOIL.lesson Quadratic Function A function that can be written in the form y = ax2 + bx + c, where a0.
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? 1) 2) 3) 4) 5) 6) 7)
y = -7x 1) Quadratic 2) Quadratic 3) Linear 4) Quadratic 5) Quadratic 6) Quadratic 7) Linear 8) Linear 9) Linear 10)Linear
Factoring Quadratic Equations How do you solve quadratic equations by factoring? 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. 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: (should be mental) - List the factor pairs whose product is 6 - Find two of those factors whose sum is 5 Ex. Factor pairs of 6: 6x1 2x3 -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. 1) x2 + 12x + 20 1) (x + 10)(x + 2) 2) x2 9x + 20
2) (x 5)(x 4) Continued Solve each equation by factoring. 1) x2 + 6x + 8 = 0 1) X = -4,-2 2) x2 2x = 3 2) X = 3, -1
3) 2x2 + 6x = -4 3) X = -1,-2 The Greatest Common Factor and Factoring by Grouping Finding the Greatest Common Factor: 1. Factor write each number in factored form. 2. List common factors 3. Choose the smallest exponents for variables and prime factors
4. Multiply the primes and variables from step 3 Always factor out the GCF first when factoring an expression The Greatest Common Factor and Factoring by Grouping Example: factor 5x2y + 25xy2z
2 1 2 1 0 5 x y 5 x y z 2 2 1 2 1 25 xy z 5 x y z 1 1 1 0
GCF 5 x y z 5 xy 2 2 5 x y 25 xy z 5 xy ( x 5 yz ) The Greatest Common Factor and Factoring by Grouping Factoring by grouping
1. Group Terms collect the terms in 2 groups that have a common factor 2. Factor within groups 3. Factor the entire polynomial factor out a common binomial factor from step 2 4. If necessary rearrange terms if step 3 didnt work, repeat steps 2 & 3 until you get 2 binomial factors The Greatest Common Factor and Factoring by Grouping Example:
10 x 2 12 y 2 15 xy 8 xy 2 2 2 ( 5 x 6 y ) work.
xy (15 8) This arrangement doesnt Rearrange and try again 10 x 2 15 xy 12 y 2 8 xy 5 x(2 x 3 y ) 4 y (3 y 2 x) (5 x 4 y )(2 x 3 y ) Factoring Trinomials of the Form x + bx + c 2
Example: Example: x 2 5x 4 4 1 5; 4 1 4 ( x 4)( x 1) x 2 4 x 21 7 3 4; 7 ( 3) 21 ( x 7)( x 3) Factoring Trinomials of the Form 2
ax + bx + c Factoring ax2 + bx + c by grouping 1. Multiply a times c 2. Find a factorization of the number from step 1 that also adds up to b 3. Split bx into these two factors multiplied by x 4. Factor by grouping (always works) Factoring Trinomials of the Form 2 ax + bx + c
Example: 8 x 2 14 x 15 ac 120 60 ( 2) 30 ( 4) 15 ( 8) 20 ( 6) b 14 20 6 Split up and factor by grouping 8 x 2 14 x 15 8 x 2 20 x 6 x 15 4 x( 2 x 5) 3( 2 x 5) ( 4 x 3)(2 x 5) Factoring Trinomials of the Form
2 ax + bx + c Factoring ax2 + bx + c by using FOIL (in reverse) 1. The first terms must give a product of ax2 (pick two) 2. The last terms must have a product of c (pick two) 3. Check to see if the sum of the outer and inner products equals bx 4. Repeat steps 1-3 until step 3 gives a sum =
bx Factoring Trinomials of the Form 2 ax + bx + c Example: 2 2 x 7 x 6 (2 x ?)( x ?) 2 try (2 x 1)( x 6) 2 x 13 x 6 incorrect 2
try (2 x 6)( x 1) 2 x 8 x 6 incorrect try (2 x 3)( x 2) 2 x 2 7 x 6 correct Factoring Trinomials of the Form 2 ax + bx + c Box Method (not in book): 2 x 2 7 x 6 (2 x ?)( x ?) 2x x 2x2 ?
? 6 Factoring Trinomials of the Form 2 ax + bx + c Box Method keep guessing until crossproduct terms add up to the middle value 2x 2 x 2x
2 4x 3 3x 6 so 2 x 2 7 x 6 (2 x 3)( x 2) Factoring Binomials and Perfect Square Trinomials Difference of 2 squares: 2 2
x y x y x y Example: 9 w2 32 w2 3 w 3 w Note: the sum of 2 squares (x2 + y2) cannot be factored. Factoring Binomials and Perfect Square Trinomials
Perfect square trinomials: 2 2 2 2 2 2
x 2 xy y x y x 2 xy y x y Examples: 2 2 2 m 6m 9 m 2 3m 3 m 3 2 2
2 2 25 z 10 z 1 5 z 2 5 z 1 5 z 1 2 Factoring Binomials and Perfect Square Trinomials Difference of 2 cubes: 3
3 2 x y x y x xy y Example: 3 3
3 2 2 w 27 w 3 w 3 w 3w 9)
Factoring Binomials and Perfect Square Trinomials Sum of 2 cubes: 3 3 2 x y x y x xy y
Example: 3 3 3 2 2
w 27 w 3 w 3 w 3w 9) Complex Numbers How are complex numbers used in solving quadratic equations? How do you add, subtract, multiply, and divide complex numbers? From previously, what if you had the equation: x2 + 25 = 0
You end up taking the of a negative number! (Calculator wont 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, youll probably only use imaginary numbers in the context of solving quadratics for their zeros. From the web Imaginary Number
i 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.
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. II. 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) III. 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 = x + 2x + 1 a=1
b=2 c=1 Discriminant: 2 - 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. Simplifying complex numbers: 1) 2) 3) 4) 5)
6) i2 = (-1)(-1) = -1 i3 i4 i5 i6 i7 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
Use distributive property to calculate: (2 + 2i)4 (2 + 2i)(2 + 2i)(2 + 2i)(2 + 2i) (4 + 8i 4)(4 + 8i 4) (8i)(8i) = 64i2 = -64 Solving Quadratic Equations by Factoring Solving a Quadratic Equation by factoring
1. Write in standard form all terms on one side of equal sign and zero on the other 2. Factor (completely) 3. Set all factors equal to zero and solve the resulting equations 4. (if time available) check your answers in the original equation Solving Quadratic Equations by Factoring Example: 2
2 x 5 7 x 2 standard form : 2 x 7 x 5 0 factored : (2 x 5)( x 1) 0 2 x 5 0 or x 1 0 solutions : x 2.5, x 1 Solve each equation by factoring. 1) x2 + 6x + 8 = 0 1) X = -4,-2
2) x2 2x = 3 2) X = 3, -1 3) 2x2 + 6x = -4 3) X = -1,-2 Properties of parabolas LEQ: How do you find the maximum or minimum of a parabola? 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 Solving by finding roots: Quadratic equations can also be solved by finding the square roots. *This method is effective when theres 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? 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 1) x2 7x = -12 1) X = 3,4 2) 6x2 = -19x 15 2) X = -1.5, -1,7 3) 5x2 7x 3 = 8 3) X = -0.9, 2.3
4) 1 = 4x2 + 3x 4) X = -1, 0.3 What if you cant factor it? If youre having trouble factoring a quadratic equation, you can always use the quadratic formula. (Graphic Organizer) Solve using any method. 1)
1) 2) 3) 4) 5) 6) X = 4, -4 X = 3, 8 X = 3.6, - 3.6 X = 3, - 0.5 X = -1.5, -0.7 X = 7, 1
T.O.T.D. Answer the LEQs. 1)When is the quadratic formula a good method for solving an equation? 2)How do you determine whether it is best to solve a quadratic equation by factoring, graphing, or using the quadratic equation? 3)How do you find the maximum or minimum of a parabola? Working Backwards The solutions to a quadratic equation are
x = -3, and x = -2. Given this information, find the given quadratic equation. Y = x2 + 5x + 6 The Discriminant (Reminder) 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. II. 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)
III. 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 = x + 2x + 1 a=1 b=2 c=1 Discriminant: 2 - 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.
Assignment Complex Numbers Quadratic Equations Quiz Warm-up 1/23/08 Solve each equation. Give an exact answer if possible. Otherwise write the answer to two decimal places.
1) x2 4x 21 = 0 2) 2x2 3x = 0 3) (x 3)(x + 4) = 12 4) (x + 1)(x 2)(2x + 1) = 0 Simplify each expression 1) -25 2) (2 + 3i)(3 4i) 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 Reasoning Before completing the square, it is useful to know what to use it for. If you know what an equation means in vertex form, it will be more meaningful. Putting an equation into vertex form can
be helpful for drawing an exact sketch for a graph. Vertex Form Y = a(x h)2 + k (h,k) is the vertex Axis of symmetry is x = h If a >0 the parabola opens narrower If 0 < a < 1 the parabola opens wider If a < 0 (negative) the parabola opens down. Ex1. Y = -(x 2)2 + 3
Vertex (2,3) Axis of symmetry x = 2 makes it open wider - Makes is open down Specific points: Ex. If x = 0, y = 1 Plot the point (0,1), the point on the other side of the axis of symmetry, the same distance away is the reflection (4,1) Now Try these Sketch each parabola. Label the vertex and the axis of symmetry.
Y = (x 2)2 3 Y = 8(x + 1)2 2 Y = 1/8(x + 1)2 1 Y= (x 6)2 + 6 Y = 4(x 1)2 2 Y = 3x2 Y = -x2 + 2 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. 1) Find the b-term.
2) Divide the b-term by 2 3) The square of this will be c. Try these: 1) x2 + 2x + ___ 2) x2 12x + ___ 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 2i5 The two solutions are:
x = 4 + 2i5 x = 4 2i5 Solve by completing the square Solve by completing the square Solve by completing the square Solve by completing the square Solve by completing the square
Re-write in vertex form Give the coordinates of the vertex Re-write in vertex form Give the coordinates of the vertex Solve using the quadratic formula Solve using the quadratic formula Evaluate the discriminant Tell how many and what solutions
Evaluate the discriminant Tell how many and what solutions Evaluate the discriminant Tell how many and what solutions Assignment *Bring Books Tomorrow* Workbook P.38 Evens only #2 -14 follow directions #32 - 40 follow directions #16 - 30
Find the following: X- intercepts (solutions) Vertex (using vertex form or min/max tool) Warm-up 1/24/08 Jan Feb Mar
Apr May Jun Jul Aug Sep
Oct Nov Dec $353 $317 $366 $284
$231 $195 $84 $62 $277 $168
$360 $322 1) Would a linear model for the heating cost over time fit these data well? Explain. 2) Would a quadratic model fit the data well? Would the model be appropriate for extrapolation? Using Calculators to Find the best quadratic model:
STATEDIT PUT DATA IN COLUMNS WINDOW: MAKE APPROPRIATE 2ND: Y=: STAT PLOT 1 (ON) GRAPH
STATCALCQUADREG ENTER 2X Another Way An easier way to find the minimum or maximum point Since parabolas are symmetric: Solve the quadratic for x (two solutions) The vertex is evenly between the two zeros, so, average your answers together This will be the x-coordinate of the vertex. To find the y-coordinate, plug in the x
Ex. Consider f(x) = 2x2 3x 2 Solve for the x-intercepts X = 2, -1/2 To find the x-coordinate of the vertex, average the answers (2 1/2)/ 2 = 0.75 Plug in 0.75 to find the y-value at this point Y = -3.125
Thus, the vertex is (0.75, -3.125) Check / Go over Homework Assignment/Exercises Textbook P.125-126 #3, 4, 5, 6 (use velocity formula on p.122), 10, 11,12a-d; 13a,b; 14a,b Warm-up
1/25/08 Given: h = -16t2 + vot + ho Ho is the initial height Vo is the initial velocity A projectile is shot from a tower 10 feet high with an upward velocity of 100 feet per second. 1) Approximate the relationship between height and time after the projectile is shot. 2) How long will the projectile be in the air?
When citing long sections (more than three lines) of poetry, keep formatting as close to the original as possible. In his poem "My Papa's Waltz," Theodore Roethke explores his childhood with his father:
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