Section 6 – Quadratics – Part 1

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Section 6 – Quadratics – 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‘₯ ! 𝑦𝑦 ! as (π‘₯π‘₯ ! )! (𝑦𝑦 ! )! , thus recognizing it as adifference of squares that can befactored (π‘₯π‘₯ ! 𝑦𝑦 ! )(π‘₯π‘₯ ! 𝑦𝑦 ! ).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‘₯ 𝑝𝑝)! π‘žπ‘ž 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 – 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 – Part 1Solving Quadratics by Factoring – Part 2Solving Quadratics by Factoring – Special CasesSolving Quadratics by Taking Square RootsSolving Quadratics by Completing the SquareQuadratics in ActionSection 6 – Video 1Real-World Examples of Quadratic FunctionsLet’s 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 – Part 1

We can represent the distance traveled in terms of time withthe equation 𝑑𝑑(𝑑𝑑) 40𝑑𝑑.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 watermelon’s 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„Ž 𝑑𝑑 16.1𝑑𝑑 ! 300Section 6: Quadratics – Part 1!119

Important facts:Why did we only consider the first quadrant of Liam’s graph?! We call this non-linear function a .! The general form (parent function) of the equation is.In Liam’s graph, what was the watermelon’s height when it hitthe ground?The graph of 𝑓𝑓(π‘₯π‘₯) π‘₯π‘₯ ! is shown below:The time when the watermelon’s 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 Liam’s 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 – Part 1ZEROS 𝑋𝑋-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 ball’s 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 function’s graph isneeded to answer this question?Part C: You want to know when the ball will reach itsmaximum height. What part of the function’s graph isneeded to answer this question?Time (in seconds)What type of function best models the rocket’s motion?After how many seconds did the rocket hit the ground?Part D: What information do you need to graph the ball’sheight 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 – 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 – 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 π‘Žπ‘Žπ‘₯π‘₯ ! 𝑏𝑏𝑏𝑏 𝑐𝑐 0, where π‘Žπ‘Ž 0,Profitπ‘₯π‘₯ 𝑏𝑏 𝑏𝑏 ! 4π‘Žπ‘Žπ‘Žπ‘Ž2π‘Žπ‘ŽTo 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 π‘Žπ‘Ž, 𝑏𝑏, and 𝑐𝑐.Part B: What do the solutions represent?3. Substitute π‘Žπ‘Ž, 𝑏𝑏, and 𝑐𝑐 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 – Part 1

Use the quadratic formula to solve 2𝑀𝑀 ! 𝑀𝑀 5.Let’s 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‘₯ 203π‘žπ‘ž! 11 20π‘žπ‘žDoes the graph verify the solutions we found using thequadratic formula?Section 6: Quadratics – Part 1Quadratic equations can always be solvedwith the quadratic formula.! There are othermethods that work, but when in doubt, usethe quadratic formula.123

Let’s use the quadratic formula to discuss the nature of thesolutions.Consider the graph of the function 𝑓𝑓 π‘₯π‘₯ π‘₯π‘₯ ! 6π‘₯π‘₯ 8.Consider the graph of the function 𝑓𝑓 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 – Part 1

Consider the graph of the function 𝑓𝑓 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 – 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 neighbor’s 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 4𝑀𝑀 ! 56𝑀𝑀 192. Use thequadratic formula to find the width, 𝑀𝑀, 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 – Part 1

Here is another multiplication problem we can express usingthe area model.Section 6 – 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‘₯ 2𝑦𝑦 7𝑧𝑧)3π‘₯π‘₯2𝑦𝑦 7𝑧𝑧(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 – Part 1127

Try It!Let’s Practice!Factor each quadratic expression.Factor each quadratic expression.2π‘₯π‘₯ ! 3π‘₯π‘₯ 5π‘Žπ‘Ž! 11π‘Žπ‘Ž 243π‘₯π‘₯ ! 8π‘₯π‘₯ 44𝑀𝑀 ! 21𝑀𝑀 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 – 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 – Video 4Solving Quadratics by Factoring – 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 π‘Žπ‘Žπ‘Žπ‘Ž 0, then either π‘Žπ‘Ž 0,  𝑏𝑏 0, Β or Β π‘Žπ‘Ž 𝑏𝑏 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 – Part 1129

Let’s Practice!Solve for 𝑏𝑏 by factoringTry it!𝑏𝑏 ! 8𝑏𝑏 15 0.Solve each quadratic equation by factoring.π‘₯π‘₯ ! 11π‘₯π‘₯ 18 0Solve for 𝑓𝑓 by factoring 10𝑓𝑓 ! 17𝑓𝑓 3 0.130Section 6: Quadratics – Part 16𝑗𝑗 ! 19𝑗𝑗 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 Tyra’swork.!Section 6: Quadratics – Part 1131

Section 6 – Video 5Solving Quadratics by Factoring – Part 2Solve for π‘šπ‘š: 3π‘šπ‘š! 30π‘šπ‘š 168 0Many quadratic equations will not be in standard form:! The equation won’t always equal zero.! There may be a greatest common factor (GCF) within allof the terms.Solve for 𝑝𝑝: 𝑝𝑝! 36 13𝑝𝑝Solve for π‘₯π‘₯: π‘₯π‘₯ 4 π‘₯π‘₯ 5 8132Section 6: Quadratics – Part 1When solving a quadratic, if the quadratic isfactored but not equal to zero, then you’vegot some work to do!

Try It!Solve for 𝑑𝑑: 6𝑑𝑑 ! 5𝑑𝑑 1BEAT THE TEST!1. What are the solutions to 40π‘₯π‘₯ ! 30π‘₯π‘₯ 135? Select all thatapply.""""! !! !! !!!"""!!!!!!Solve for 𝑦𝑦: 200𝑦𝑦 ! 900𝑦𝑦 1000!Section 6: Quadratics – 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 – Video 6Solving Quadratics by Factoring – 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 π‘Žπ‘Ž! 2π‘Žπ‘Žπ‘Žπ‘Ž 𝑏𝑏 ! .134Section 6: Quadratics – Part 1

Factoring a Perfect Square Trinomial! If π‘Žπ‘Ž! 2π‘Žπ‘Žπ‘Žπ‘Ž 𝑏𝑏 ! is a perfect square trinomial, thenπ‘Žπ‘Ž! 2π‘Žπ‘Žπ‘Žπ‘Ž 𝑏𝑏 ! π‘Žπ‘Ž 𝑏𝑏 ! .! If π‘Žπ‘Ž! 2π‘Žπ‘Žπ‘Žπ‘Ž 𝑏𝑏 ! is a perfect square trinomial, thenπ‘Žπ‘Ž! 2π‘Žπ‘Žπ‘Žπ‘Ž 𝑏𝑏 ! π‘Žπ‘Ž 𝑏𝑏 ! .Let’s Practice!Determine whether the following expressions are perfectsquare trinomials.Solve for π‘žπ‘ž: π‘žπ‘ž! 10π‘žπ‘ž 25 0Solve for 𝑀𝑀: 4𝑀𝑀 ! 49 28𝑀𝑀16π‘₯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 – Part 1135

Try It!Let’s Practice!Solve for π‘₯π‘₯: π‘₯π‘₯ ! 16 0Solve the equation 49π‘˜π‘˜ ! 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 121𝑝𝑝! 100(2π‘₯π‘₯ 7)(2π‘₯π‘₯ 7)(5π‘₯π‘₯ 1)(5π‘₯π‘₯ 1)𝑝𝑝! 144 0Describe any patterns you notice.! When we have a binomial in the form π‘Žπ‘Ž! 𝑏𝑏 ! , it is calledthe difference of two squares. We can factor this asπ‘Žπ‘Ž 𝑏𝑏 π‘Žπ‘Ž 𝑏𝑏 .136Section 6: Quadratics – Part 1

BEAT THE TEST!1. Which of the following expressions are equivalent to8π‘Žπ‘Ž! 98π‘Žπ‘Ž? Select all that apply.""""""2(4π‘Žπ‘Ž! 49π‘Žπ‘Ž)2π‘Žπ‘Ž(4π‘Žπ‘Ž! 49)2π‘Žπ‘Ž(4π‘Žπ‘Ž! 49π‘Žπ‘Ž)2π‘Žπ‘Ž 7 (2π‘Žπ‘Ž 7)2 2π‘Žπ‘Ž 7 (2π‘Žπ‘Ž 7)2π‘Žπ‘Ž 2π‘Žπ‘Ž 7 (2π‘Žπ‘Ž 7)Section 6 – 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 π‘Žπ‘Žπ‘Žπ‘Ž ! 𝑐𝑐 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.! Don’t forget the negative root!2. A bird flies to the ground and lands, only to be scared awayby a cat. The bird’s motion can be described by theequation β„Ž 𝑑𝑑 4𝑑𝑑 ! 20𝑑𝑑 25, where 𝑑𝑑 represents time inseconds and β„Ž(𝑑𝑑) represents the bird’s 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 – 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„Ž(𝑑𝑑) 16𝑑𝑑 Β ! 𝑠𝑠, where β„Ž(𝑑𝑑)represents the lifebelt’s height from the sea at any giventime, 𝑑𝑑 is the time in seconds, and 𝑠𝑠 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 – Part 1

Section 6 – Video 8Solving Quadratics by Completing the SquareSometimes, you won’t 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 – Part 1139

Let’s see how this can be used to solve quadratic equations.Try It!Recall from a previous video how we factored perfect squaretrinomials. If π‘Žπ‘Žπ‘₯π‘₯ ! 𝑏𝑏𝑏𝑏 𝑐𝑐 is a perfect square trinomial, thenComplete the square to solve the following equations.π‘Žπ‘Žπ‘₯π‘₯ ! 𝑏𝑏𝑏𝑏 𝑐𝑐 π‘₯π‘₯ ! 6π‘₯π‘₯ 0π‘Žπ‘Žπ‘₯π‘₯ 𝑐𝑐!and π‘Žπ‘Žπ‘₯π‘₯ ! 𝑏𝑏𝑏𝑏 𝑐𝑐 !π‘Žπ‘Žπ‘₯π‘₯ 𝑐𝑐 .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 – 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 π‘Žπ‘Ž does not equal one, divide every term in the equationby π‘Žπ‘Ž.3. Subtract 𝑐𝑐 from both sides.4. Divide 𝑏𝑏 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 – Part 1141

Section 6 – Video 9Quadratics in ActionQuestionLet’s 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 𝑦𝑦-intercept. In thestandard form, π‘Žπ‘Žπ‘₯π‘₯ ! 𝑏𝑏𝑏𝑏 𝑐𝑐, 𝑐𝑐 Β is the𝑦𝑦-intercept.This is the π‘₯π‘₯-coordinate of the vertex,π‘₯π‘₯ !!!!, where values of π‘Žπ‘Ž and 𝑏𝑏come 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 – Part 1This is the 𝑦𝑦-coordinate of the vertex.Plug in the π‘₯π‘₯-coordinate from thestep above and evaluate to find 𝑦𝑦. Invertex form, the height is π‘˜π‘˜ and thevertex is (β„Ž, Β π‘˜π‘˜).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‘₯- ‐intercept(s).In function 𝐻𝐻 𝑑𝑑 π‘Žπ‘Žπ‘‘π‘‘ ! 𝑏𝑏𝑏𝑏 𝑐𝑐, ifheight is given, then substitute thevalue for 𝐻𝐻(𝑑𝑑). If time is given, thensubstitute for 𝑑𝑑.

Let’s Practice!What is the maximum height reached by the javelin?An athlete throws a javelin. The javelin’s height above theground, in feet, after it has traveled a horizontal distance for 𝑑𝑑seconds is given by the equation:β„Ž 𝑑𝑑 Β  0.08𝑑𝑑 ! 0.64𝑑𝑑 5.15What would the graph of β„Ž(𝑑𝑑) versus 𝑑𝑑 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 – 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„Ž 𝑑𝑑 16𝑑𝑑 ! 80𝑑𝑑, whereβ„Ž(𝑑𝑑) is the height of the ball, in feet, and 𝑑𝑑 is the time inseconds it has been in the air.What would the graph of β„Ž(𝑑𝑑) versus 𝑑𝑑 look like?When is the ball 65 feet in the air? Explain.Why is the 𝑦𝑦-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 – 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 β€œU-shaped parabolas.” In these parabolas,look for minimums rather than maximums.How high was the seagull flying before he dove down to takethe Pak family’s food?Let’s 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„Ž 𝑑𝑑 Β  (𝑑𝑑 7)!!Section 6: Quadratics – Part 1145

Part C: What is the bottle rocket’s 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„Ž 𝑑𝑑 0.5π‘Žπ‘Žπ‘‘π‘‘ ! 𝑣𝑣𝑣𝑣 𝑠𝑠, where the gravitationalconstant, π‘Žπ‘Ž, is 32 feet per square second, 𝑣𝑣 is the initialvelocity, 𝑠𝑠 is the initial height, and β„Ž(𝑑𝑑) is the height in feetmodeled as a function of time, 𝑑𝑑.Part A: What function describes the height, β„Ž, of the bottlerocket after 𝑑𝑑 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 – Part 1

Nov 01, 2015Β Β· Solving Quadratics by Factoring – 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

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