9m ago

7 Views

1 Downloads

1.72 MB

30 Pages

Transcription

Section 6 β Quadratics β Part 1MAFS.912.A-REI.2.4The following Mathematics Florida Standards will becovered in this 12.F-IF.3.8Use the structure of an expressionto identify ways to rewrite it. Forexample, see π₯π₯ ! π¦π¦ ! as (π₯π₯ ! )! (π¦π¦ ! )! , thus recognizing it as adifference of squares that can befactored (π₯π₯ ! π¦π¦ ! )(π₯π₯ ! π¦π¦ ! ).Choose and produce anequivalent form of an expression toreveal and explain properties ofthe quantity represented by theexpression.a. Factor a quadratic expressionto reveal the zeros of thefunction it defines.Write a function defined by anexpression in different butequivalent forms to reveal andexplain different properties of thefunction.a. Use the process of factoringand completing the square inquadratic function to showzeros, extreme values, andsymmetry of the graph, andinterpret these in terms of acontext.MAFS.912.F-IF.2.4Solve quadratic equations in onevariable.a. Use the method of completingthe square to transform anyquadratic equation in π₯π₯ Β into anequation of the form (π₯π₯ ππ)! ππ Β that has the same solutions.Derive the quadratic formulafrom this form.b. Solve quadratic equations byinspection (e.g., for π₯π₯ ! 49),taking square roots,completing the square, thequadratic formula, andfactoring, as appropriate tothe initial form of the equation.Recognize when the quadraticformula gives complexsolutions.For a function that models arelationship between twoquantities, interpret key features ofgraphs and tables in terms of thequantities and sketch graphsshowing key features given averbal description of therelationship. Key features include:intercepts; intervals where thefunction is increasing, decreasing,positive, or negative; relativemaximums and minimums;symmetries; end behavior; andperiodicity.!Section 6: Quadratics β Part 1117

Videos in this SectionVideo 1:Video 2:Video 3:Video 4:Video 5:Video 6:Video 7:Video 8:Video 9:Real-World Examples of Quadratic FunctionsSolving Quadratics Using the Quadratic FormulaFactoring Quadratic ExpressionsSolving Quadratics by Factoring β Part 1Solving Quadratics by Factoring β Part 2Solving Quadratics by Factoring β Special CasesSolving Quadratics by Taking Square RootsSolving Quadratics by Completing the SquareQuadratics in ActionSection 6 β Video 1Real-World Examples of Quadratic FunctionsLetβs revisit linear functions.Imagine that you are driving down the road at a constantspeed of 40 Β mph. This is a linear function.We can represent the distance traveled versus time on atable:Time(in hours)1234DistanceTraveled(in miles)4080120160We can represent the scenario on a graph:Distance (in miles)Distance TraveledTime (in hours)118Section 6: Quadratics β Part 1

We can represent the distance traveled in terms of time withthe equation ππ(π‘π‘) 40π‘π‘.Liam entered the data into his graphing calculator. The graphbelow displays the first quadrant of the graph.Height (in feet)Linear functions always have a constant rate of change. In thissection, we are going to discover a type of non-linearfunction.Consider the following situation:Liam dropped a watermelon from the top of a 300 feet tallbuilding. He wanted to know if the watermelon was falling at aconstant rate over time. He filmed the watermelonβs fall andthen recorded his observations in the following table:Time(in seconds)0Height(in feet)3003155.1124Time (in seconds)283.9What is the independent variable?235.642.4What do you notice about the rate of change?What is the dependent variable?Why do you think that the rate of change is not constant?Liam then used his calculator to find the equation of thefunction:β π‘π‘ 16.1π‘π‘ ! 300Section 6: Quadratics β Part 1!119

Important facts:Why did we only consider the first quadrant of Liamβs graph?! We call this non-linear function a .! The general form (parent function) of the equation is.In Liamβs graph, what was the watermelonβs height when it hitthe ground?The graph of ππ(π₯π₯) π₯π₯ ! is shown below:The time when the watermelonβs height was at zero is calledthe solution to this quadratic equation. We also call these thezeros of the equation.There was only one solution to Liamβs equation. Describe asituation where there could be two solutions.What about no solutions?To solve a quadratic equation using a graph:! look for the of the graph! the solution(s) are the values where the graph interceptsthe! This graph is called a .120Section 6: Quadratics β Part 1ZEROS ππ-INTERCEPTS SOLUTIONS

Try It!BEAT THE TEST!What are the solutions to the quadratic equation graphedbelow?1. A ball is thrown straight up into the air.Part A: What type of function best models the ballβs heightover time? What would it look like?Height (in meters)Aaron shoots a water bottle rocket from the ground. A graphof height over time is shown below:Part B: You want to know how long it will take until the ballhits the ground. What part of the functionβs graph isneeded to answer this question?Part C: You want to know when the ball will reach itsmaximum height. What part of the functionβs graph isneeded to answer this question?Time (in seconds)What type of function best models the rocketβs motion?After how many seconds did the rocket hit the ground?Part D: What information do you need to graph the ballβsheight over time?Estimate the maximum height of the rocket.!This point is called the vertex, the maximum or minimum pointof a parabola.Section 6: Quadratics β Part 1121

2. Jordan owns an electronics business. During her first year inthe business, she collected data on different prices thatyielded different profits. She used the data to create thefollowing graph showing the relationship between the sellingprice of an item and the profit:Electronics SalesSection 6 β Video 2Solving Quadratics Using the Quadratic FormulaHow can we find the solutions when the quadratic is given inthe form of an equation?We can always use the quadratic formula.For any quadratic equation πππ₯π₯ ! ππππ ππ 0, where ππ 0,Profitπ₯π₯ ππ ππ ! 4ππππ2ππTo use the quadratic formula:1. Set the quadratic equation equal to zero.Selling PricePart A: Circle the solutions to the quadratic functiongraphed above.2. Identify ππ, ππ, and ππ.Part B: What do the solutions represent?3. Substitute ππ, ππ, and ππ into the quadratic formula andevaluate to find the zeros.Part C: Box the vertex of the graph.Part D: What does the vertex represent?122Section 6: Quadratics β Part 1

Use the quadratic formula to solve 2π€π€ ! π€π€ 5.Letβs Practice!Use the quadratic formula to solve π₯π₯ ! 4π₯π₯ 3 0.Try It!Use the quadratic formula to solve the following quadraticequations.We can always verify our answers with a graph.Consider the graph of the quadratic equation π₯π₯ ! 4π₯π₯ 3 0.π₯π₯ ! π₯π₯ 203ππ! 11 20ππDoes the graph verify the solutions we found using thequadratic formula?Section 6: Quadratics β Part 1Quadratic equations can always be solvedwith the quadratic formula.! There are othermethods that work, but when in doubt, usethe quadratic formula.123

Letβs use the quadratic formula to discuss the nature of thesolutions.Consider the graph of the function ππ π₯π₯ π₯π₯ ! 6π₯π₯ 8.Consider the graph of the function ππ π₯π₯ π₯π₯ ! 4π₯π₯ 4.Where does this parabola intersect the π₯π₯-axis?Where does this parabola intersect the π₯π₯-axis?Use the quadratic formula to find the zero(s) of the function.Use the quadratic formula to find the zero(s) of the function.124Section 6: Quadratics β Part 1

Consider the graph of the function ππ π₯π₯ π₯π₯ ! 6π₯π₯ 11.When using the quadratic formula, if thediscriminant of the quadratic (part under theradical) results in a negative number thensolutions are non-real, complex solutions.Try It!Determine if the following quadratic equations have complexor real solution(s).2π₯π₯ ! 3π₯π₯ 10 0Where does this parabola intersect the π₯π₯-axis?π₯π₯ ! 6π₯π₯ 9 0Use the quadratic formula to find the zero(s) of the function.π₯π₯ ! 8π₯π₯ 20 0!Section 6: Quadratics β Part 1125

BEAT THE TEST!1. Which of the following quadratic equations have realsolutions? Select all that apply.""""" 3π₯π₯ ! 5π₯π₯ 11 π₯π₯ ! 12π₯π₯ 6 02π₯π₯ ! π₯π₯ 6 05π₯π₯ ! 10π₯π₯ 3π₯π₯ ! 2π₯π₯ 82. Your neighborβs garden measures 12 meters by 16 meters.He plans to install a pedestrian pathway all around it,increasing the total area to 285 square meters. The newarea can be represented by 4π€π€ ! 56π€π€ 192. Use thequadratic formula to find the width, π€π€, of the pathway.Part A: Write an equation that can be used to solve for thewidth of the pathway.Part B: Use the quadratic formula to solve for the width ofthe pathway.126Section 6: Quadratics β Part 1

Here is another multiplication problem we can express usingthe area model.Section 6 β Video 3Factoring Quadratic ExpressionsWe can factor quadratic expressions by using the samedistributive property that we practiced in Section 1.Think back to the area model we used for the distributiveproperty:3(π₯π₯ 2π¦π¦ 7π§π§)3π₯π₯2π¦π¦ 7π§π§(2π₯π₯ 5)(π₯π₯ 3)2π₯π₯ 5π₯π₯ 3We can use this same model with the distributive property tofactor a quadratic expression. Notice the following fourpatterns:! The first term of the trinomial can always be found in therectangle.! The last term of the trinomial can always be found in therectangle.We can also use the area model with the distributive propertyto factor out the greatest common factor (GCF) of anexpression.10π₯π₯ ! 14π₯π₯ ! 12π₯π₯10π₯π₯ ! 14π₯π₯ !! The second term of the trinomial is the of theandrectangles.! The of the is always equal. 12xFor example:2π₯π₯ ! 15 30π₯π₯ !5π₯π₯ 6π₯π₯ 30π₯π₯ !We can use these four patterns to factor any quadraticexpression.!Section 6: Quadratics β Part 1127

Try It!Letβs Practice!Factor each quadratic expression.Factor each quadratic expression.2π₯π₯ ! 3π₯π₯ 5ππ! 11ππ 243π₯π₯ ! 8π₯π₯ 44π€π€ ! 21π€π€ 20128You can check your answer to every factorby using the distributive property. The productof the factors should always result in thetrinomial that you started with (your originalproblem).Section 6: Quadratics β Part 1

BEAT THE TEST!1. Identify all factors of the expression 18π₯π₯ ! 9π₯π₯ 5."""""2π₯π₯ 56π₯π₯ 518π₯π₯ 53π₯π₯ 53π₯π₯ 1Section 6 β Video 4Solving Quadratics by Factoring β Part 1Solve a quadratic equation by factoring:! Once a quadratic equation is factored, we can use thezero product property to solve the equation.! The zero product property states that if the product of twofactors is zero, then one (or both) of the factors must beo If ππππ 0, then either ππ 0, Β ππ 0, Β or Β ππ ππ 0.To solve a quadratic equation by factoring:1. Set the equation equal to zero.2. Factor the quadratic.3. Set each factor equal to zero and solve.4. Write the solution set.!Section 6: Quadratics β Part 1129

Letβs Practice!Solve for ππ by factoringTry it!ππ ! 8ππ 15 0.Solve each quadratic equation by factoring.π₯π₯ ! 11π₯π₯ 18 0Solve for ππ by factoring 10ππ ! 17ππ 3 0.130Section 6: Quadratics β Part 16ππ ! 19ππ 14 0

BEAT THE TEST!1. Which of the following quadratic equations has the solution!set , Β 6 Β ?""""""!(π₯π₯ 6)(3π₯π₯ 5) 0(3π₯π₯ 5)(π₯π₯ 6) 0(5π₯π₯ 3)(π₯π₯ 6) 0(5π₯π₯ 3)(π₯π₯ 6) 03π₯π₯ 5 2π₯π₯ 12 0( 3π₯π₯ 5)(π₯π₯ 6) 02. Tyra solved the quadratic equation π₯π₯ ! 10π₯π₯ 24 0 byfactoring. Her work is shown below:Step 1:π₯π₯ ! 10π₯π₯ 24 0Step 2: Β Β Β Β Β Β Β π₯π₯ ! 4π₯π₯ 6π₯π₯ 24 0π₯π₯ ! 4π₯π₯ ( 6π₯π₯ 24) 0Step 3:Step 4:π₯π₯ π₯π₯ 4 6(π₯π₯ 4) 0Step 5:(π₯π₯ 4)(π₯π₯ 6) 0Step 6:π₯π₯ 4 0, Β π₯π₯ 6 0Step 7:π₯π₯ 4 Β or Β π₯π₯ 64, Β 6Step 8:Tyra did not find the correct solutions. Identify the step(s)where she made mistakes and explain how to correct Tyraβswork.!Section 6: Quadratics β Part 1131

Section 6 β Video 5Solving Quadratics by Factoring β Part 2Solve for ππ: 3ππ! 30ππ 168 0Many quadratic equations will not be in standard form:! The equation wonβt always equal zero.! There may be a greatest common factor (GCF) within allof the terms.Solve for ππ: ππ! 36 13ππSolve for π₯π₯: π₯π₯ 4 π₯π₯ 5 8132Section 6: Quadratics β Part 1When solving a quadratic, if the quadratic isfactored but not equal to zero, then youβvegot some work to do!

Try It!Solve for ππ: 6ππ ! 5ππ 1BEAT THE TEST!1. What are the solutions to 40π₯π₯ ! 30π₯π₯ 135? Select all thatapply.""""! !! !! !!!"""!!!!!!Solve for π¦π¦: 200π¦π¦ ! 900π¦π¦ 1000!Section 6: Quadratics β Part 1133

2. The area of the rectangle is 105 Β square units.What is the value of π₯π₯?2π₯π₯ 1π₯π₯Section 6 β Video 6Solving Quadratics by Factoring β Special CasesThere are a few special cases when solving quadratics byfactoring.Perfect Square Trinomials! π₯π₯ ! 6π₯π₯ 9 is an example of perfect square trinomial. Wesee this when we factor.! A perfect square trinomial is created when you square a.Recognizing a Perfect Square TrinomialA quadratic expression can be factored as a perfect squaretrinomial if it can be re-written in the form ππ! 2ππππ ππ ! .134Section 6: Quadratics β Part 1

Factoring a Perfect Square Trinomial! If ππ! 2ππππ ππ ! is a perfect square trinomial, thenππ! 2ππππ ππ ! ππ ππ ! .! If ππ! 2ππππ ππ ! is a perfect square trinomial, thenππ! 2ππππ ππ ! ππ ππ ! .Letβs Practice!Determine whether the following expressions are perfectsquare trinomials.Solve for ππ: ππ! 10ππ 25 0Solve for π€π€: 4π€π€ ! 49 28π€π€16π₯π₯ ! 44π₯π₯ 121What do you notice about the number of solutions to perfectsquare quadratic equations?Sketch the graph of a quadratic equation that is a perfectsquare trinomial.π₯π₯ ! 8π₯π₯ 64!Section 6: Quadratics β Part 1135

Try It!Letβs Practice!Solve for π₯π₯: π₯π₯ ! 16 0Solve the equation 49ππ ! 64.Difference of SquaresUse the distributive property to multiply the following binomials.Try It!(π₯π₯ 5)(π₯π₯ 5)Solve each quadratic equation.0 121ππ! 100(2π₯π₯ 7)(2π₯π₯ 7)(5π₯π₯ 1)(5π₯π₯ 1)ππ! 144 0Describe any patterns you notice.! When we have a binomial in the form ππ! ππ ! , it is calledthe difference of two squares. We can factor this asππ ππ ππ ππ .136Section 6: Quadratics β Part 1

BEAT THE TEST!1. Which of the following expressions are equivalent to8ππ! 98ππ? Select all that apply.""""""2(4ππ! 49ππ)2ππ(4ππ! 49)2ππ(4ππ! 49ππ)2ππ 7 (2ππ 7)2 2ππ 7 (2ππ 7)2ππ 2ππ 7 (2ππ 7)Section 6 β Video 7Solving Quadratics by Taking Square RootsHow would you solve a quadratic equation like the onebelow?2π₯π₯ ! 36 0When quadratic equations are in the form ππππ ! ππ 0, solve bytaking the square root.1. Get the variable on the left and the constant on the right.2. Then take the square root of both sides of the equation.! Donβt forget the negative root!2. A bird flies to the ground and lands, only to be scared awayby a cat. The birdβs motion can be described by theequation β π‘π‘ 4π‘π‘ ! 20π‘π‘ 25, where π‘π‘ represents time inseconds and β(π‘π‘) represents the birdβs height.Part A: At what time will the bird land on the ground?Solve for π₯π₯ by taking the square root.2π₯π₯ ! 36 0Part B: What ordered pair represents your solution?!Section 6: Quadratics β Part 1137

Try It!Solve π₯π₯ ! 121 0.BEAT THE TEST!1. What is the smallest solution to the equation2π₯π₯ ! 17 179?ABCDSolve 5π₯π₯ ! 80 0. 9 3392. A rescuer on a helicopter that is 50 feet above the sea dropsa lifebelt. The distance from the lifebelt to the sea can bemodeled by the equation β(π‘π‘) 16π‘π‘ Β ! π π , where β(π‘π‘)represents the lifebeltβs height from the sea at any giventime, π‘π‘ is the time in seconds, and π π is the initial height fromthe sea, in feet.How long will it take for the lifebelt to reach the sea? Roundyour answer to the nearest tenth of a second.seconds138Section 6: Quadratics β Part 1

Section 6 β Video 8Solving Quadratics by Completing the SquareSometimes, you wonβt be able to solve a quadratic equationby factoring. However, you can rewrite the quadraticequation so that you can complete the square to factor andsolve.Try It!What value could be added to the quadratic to make it aperfect square trinomial?π₯π₯ ! 14π₯π₯ What value could be added to the quadratic to make it aperfect square trinomial?π₯π₯ ! 6π₯π₯ π₯π₯ ! 30π₯π₯ π₯π₯ ! 18π₯π₯ π₯π₯ ! 22π₯π₯ 71 π₯π₯ ! 8π₯π₯ 3 π₯π₯ ! 4π₯π₯ 57 π₯π₯ ! 10π₯π₯ 32 !Section 6: Quadratics β Part 1139

Letβs see how this can be used to solve quadratic equations.Try It!Recall from a previous video how we factored perfect squaretrinomials. If πππ₯π₯ ! ππππ ππ is a perfect square trinomial, thenComplete the square to solve the following equations.πππ₯π₯ ! ππππ ππ π₯π₯ ! 6π₯π₯ 0πππ₯π₯ ππ!and πππ₯π₯ ! ππππ ππ !πππ₯π₯ ππ .π₯π₯ ! 16π₯π₯ 0π₯π₯ ! 6π₯π₯ 0 3π₯π₯ ! 6π₯π₯ 02π₯π₯ ! 6π₯π₯ 6140 5π₯π₯ ! 30π₯π₯ 02π₯π₯ ! 4π₯π₯ 3Section 6: Quadratics β Part 1

To summarize, here are the steps for solving a quadratic bycompleting the square:BEAT THE TEST!1. Demonstrate how to solve 2π₯π₯ ! 24π₯π₯ 29 0 by completingthe square. Place the equations in the correct order.1. Write the equation in standard form.2. If ππ does not equal one, divide every term in the equationby ππ.3. Subtract ππ from both sides.4. Divide ππ by two and square the result. Add this value toboth sides of the equation to create a perfect squaretrinomial.5. Rewrite the equation as a perfect square trinomial.A) π₯π₯ ! 12π₯π₯ 36 14.5 36E) π₯π₯ 6C) π₯π₯ ! 12π₯π₯ 14.5G)B) π₯π₯ 6 50.5D) π₯π₯ 6 50.5! 50.5F) π₯π₯ ! 12π₯π₯ 14.5 0π₯π₯ 6! 50.56. Factor the trinomial.7. Take the square root of both sides.8. Solve for π₯π₯.!Section 6: Quadratics β Part 1141

Section 6 β Video 9Quadratics in ActionQuestionLetβs consider solving some real-world situations that involvequadratic functions.1.From what height wasthe object launched?Consider an object being launched into the air. We comparethe height versus time elapsed. Consider these questions:! From what height wasthe object launched?Height (in meters)! How long does it takethe object to reach itsmaximum height?How long did it take2. the object to reach itsmaximum height?This is typically the π¦π¦-intercept. In thestandard form, πππ₯π₯ ! ππππ ππ, ππ Β is theπ¦π¦-intercept.This is the π₯π₯-coordinate of the vertex,π₯π₯ !!!!, where values of ππ and ππcome from the standard form of aquadratic equation. π₯π₯ !!!!is also theequation that represents the axis ofsymmetry.Time (in seconds)3.What was themaximum height?! What is the maximum height?! How long does it take until the object hits the ground?At what time(s) was4. the object on theground?! At what time will the object reach a certain height or howhigh will the object be after a certain time?At what time did theobject reach a certain5. height or how high wasthe object after acertain time?142How to Answer itSection 6: Quadratics β Part 1This is the π¦π¦-coordinate of the vertex.Plug in the π₯π₯-coordinate from thestep above and evaluate to find π¦π¦. Invertex form, the height is ππ and thevertex is (β, Β ππ).The π₯π₯-intercept(s) are the solution(s),or zero(s), of the quadratic function.Solve by factoring, using thequadratic formula, or by completingthe square. In a graph, look at theπ₯π₯- βintercept(s).In function π»π» π‘π‘ πππ‘π‘ ! ππππ ππ, ifheight is given, then substitute thevalue for π»π»(π‘π‘). If time is given, thensubstitute for π‘π‘.

Letβs Practice!What is the maximum height reached by the javelin?An athlete throws a javelin. The javelinβs height above theground, in feet, after it has traveled a horizontal distance for π‘π‘seconds is given by the equation:β π‘π‘ Β 0.08π‘π‘ ! 0.64π‘π‘ 5.15What would the graph of β(π‘π‘) versus π‘π‘ look like?From what height was the javelin thrown?How high is the javelin three seconds after it was launched?When is the javelin three feet above the ground?How long does it take until the javelin hits the ground?When did the javelin reach its maximum height?!Section 6: Quadratics β Part 1143

Try It!What is the height of the ball at 3.5 seconds? When is the ballat the same height?Ferdinand is playing golf. He hits a shot off the tee box that hasa height modeled by the function β π‘π‘ 16π‘π‘ ! 80π‘π‘, whereβ(π‘π‘) is the height of the ball, in feet, and π‘π‘ is the time inseconds it has been in the air.What would the graph of β(π‘π‘) versus π‘π‘ look like?When is the ball 65 feet in the air? Explain.Why is the π¦π¦-intercept at the origin?How long does it take until the golf ball hits the ground?When does the ball reach its maximum height? What is themaximum height of the ball?144Section 6: Quadratics β Part 1

Suppose Ferdinand hits a second ball from a tee box that waselevated eight feet above the fairway. What effect does thishave on the function? Write a function that describes the newpath of the ball. Compare and contrast both functions.Describe the graph of height versus time.These are called βU-shaped parabolas.β In these parabolas,look for minimums rather than maximums.How high was the seagull flying before he dove down to takethe Pak familyβs food?Letβs Practice!The height of a seagull over time as it bobs up and down overthe ocean has a shape, or trajectory, of a parabola ormultiple parabolas.What is the minimum height of the seagull? How much timedid it take the seagull to dive down for the food?The Pak family was enjoying a great day at the beach. Atlunchtime, they took out food. A seagull swooped down,grabbed some of the food, and flew back up again. Its heightabove the ground, in meters, after it has traveled a horizontaldistance for t seconds is given by the function:What is the height of the seagull after nine seconds? Describethe scenario.β π‘π‘ Β (π‘π‘ 7)!!Section 6: Quadratics β Part 1145

Part C: What is the bottle rocketβs maximum height?BEAT THE TEST!1. Baymeadows Pointe is throwing a huge fireworkscelebration for the 4th of July. The president of theneighborhood association launched a bottle rocket upwardfrom the ground with an initial velocity of 160 feet persecond. Consider the formula for vertical motion of anobject: β π‘π‘ 0.5πππ‘π‘ ! π£π£π£π£ π π , where the gravitationalconstant, ππ, is 32 feet per square second, π£π£ is the initialvelocity, π π is the initial height, and β(π‘π‘) is the height in feetmodeled as a function of time, π‘π‘.Part A: What function describes the height, β, of the bottlerocket after π‘π‘ seconds have elapsed?Part D: What is the height of the bottle rocket after threeseconds? When is it at this height again?Part B: Sketch a graph of the height of the bottle rocket as afunction of time, and give a written description of thegraph.Part E: Suppose the bottle rocket is launched from the topof a 200-foot-tall building. How does this change theheight versus time function for the bottle rocket?What does the new graph tell you about thesituation?146Section 6: Quadratics β Part 1

Nov 01, 2015Β Β· Solving Quadratics by Factoring β Part 1 Solve a quadratic equation by factoring: Once a quadratic equation is factored, we can use the zero product property to solve the equation. ! The zero product property states that if the product of two factors is

Related Documents: