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 𝑥𝑥 ! 𝑦𝑦 ! as (𝑥𝑥 ! )! (𝑦𝑦 ! )! , thus recognizing it as adifference of squares that can befactored (𝑥𝑥 ! 𝑦𝑦 ! )(𝑥𝑥 ! 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 𝑥𝑥 into anequation of the form (𝑥𝑥 𝑝𝑝)! 𝑞𝑞 that has the same solutions.Derive the quadratic formulafrom this form.b. Solve quadratic equations byinspection (e.g., for 𝑥𝑥 ! 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 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(𝑡𝑡) 40𝑡𝑡.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:ℎ 𝑡𝑡 16.1𝑡𝑡 ! 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(𝑥𝑥) 𝑥𝑥 ! 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 𝑎𝑎𝑥𝑥 ! 𝑏𝑏𝑏𝑏 𝑐𝑐 0, where 𝑎𝑎 0,Profit𝑥𝑥 𝑏𝑏 𝑏𝑏 ! 4𝑎𝑎𝑎𝑎2𝑎𝑎To 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 𝑎𝑎, 𝑏𝑏, and 𝑐𝑐.Part B: What do the solutions represent?3. Substitute 𝑎𝑎, 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 2𝑤𝑤 ! 𝑤𝑤 5.Let’s Practice!Use the quadratic formula to solve 𝑥𝑥 ! 4𝑥𝑥 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 𝑥𝑥 ! 4𝑥𝑥 3 0.𝑥𝑥 ! 𝑥𝑥 203𝑞𝑞! 11 20𝑞𝑞Does 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 𝑥𝑥 𝑥𝑥 ! 6𝑥𝑥 8.Consider the graph of the function 𝑓𝑓 𝑥𝑥 𝑥𝑥 ! 4𝑥𝑥 4.Where does this parabola intersect the 𝑥𝑥-axis?Where does this parabola intersect the 𝑥𝑥-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 𝑥𝑥 𝑥𝑥 ! 6𝑥𝑥 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).2𝑥𝑥 ! 3𝑥𝑥 10 0Where does this parabola intersect the 𝑥𝑥-axis?𝑥𝑥 ! 6𝑥𝑥 9 0Use the quadratic formula to find the zero(s) of the function.𝑥𝑥 ! 8𝑥𝑥 20 0!Section 6: Quadratics – Part 1125
BEAT THE TEST!1. Which of the following quadratic equations have realsolutions? Select all that apply.""""" 3𝑥𝑥 ! 5𝑥𝑥 11 𝑥𝑥 ! 12𝑥𝑥 6 02𝑥𝑥 ! 𝑥𝑥 6 05𝑥𝑥 ! 10𝑥𝑥 3𝑥𝑥 ! 2𝑥𝑥 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 4𝑤𝑤 ! 56𝑤𝑤 192. Use thequadratic formula to find the width, 𝑤𝑤, 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(𝑥𝑥 2𝑦𝑦 7𝑧𝑧)3𝑥𝑥2𝑦𝑦 7𝑧𝑧(2𝑥𝑥 5)(𝑥𝑥 3)2𝑥𝑥 5𝑥𝑥 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.10𝑥𝑥 ! 14𝑥𝑥 ! 12𝑥𝑥10𝑥𝑥 ! 14𝑥𝑥 !! The second term of the trinomial is the of theandrectangles.! The of the is always equal. 12xFor example:2𝑥𝑥 ! 15 30𝑥𝑥 !5𝑥𝑥 6𝑥𝑥 30𝑥𝑥 !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.2𝑥𝑥 ! 3𝑥𝑥 5𝑎𝑎! 11𝑎𝑎 243𝑥𝑥 ! 8𝑥𝑥 44𝑤𝑤 ! 21𝑤𝑤 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 18𝑥𝑥 ! 9𝑥𝑥 5."""""2𝑥𝑥 56𝑥𝑥 518𝑥𝑥 53𝑥𝑥 53𝑥𝑥 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 𝑎𝑎𝑎𝑎 0, then either 𝑎𝑎 0, 𝑏𝑏 0, or 𝑎𝑎 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.𝑥𝑥 ! 11𝑥𝑥 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 , 6 ?""""""!(𝑥𝑥 6)(3𝑥𝑥 5) 0(3𝑥𝑥 5)(𝑥𝑥 6) 0(5𝑥𝑥 3)(𝑥𝑥 6) 0(5𝑥𝑥 3)(𝑥𝑥 6) 03𝑥𝑥 5 2𝑥𝑥 12 0( 3𝑥𝑥 5)(𝑥𝑥 6) 02. Tyra solved the quadratic equation 𝑥𝑥 ! 10𝑥𝑥 24 0 byfactoring. Her work is shown below:Step 1:𝑥𝑥 ! 10𝑥𝑥 24 0Step 2: 𝑥𝑥 ! 4𝑥𝑥 6𝑥𝑥 24 0𝑥𝑥 ! 4𝑥𝑥 ( 6𝑥𝑥 24) 0Step 3:Step 4:𝑥𝑥 𝑥𝑥 4 6(𝑥𝑥 4) 0Step 5:(𝑥𝑥 4)(𝑥𝑥 6) 0Step 6:𝑥𝑥 4 0, 𝑥𝑥 6 0Step 7:𝑥𝑥 4 or 𝑥𝑥 64, 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 𝑚𝑚: 3𝑚𝑚! 30𝑚𝑚 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 𝑥𝑥: 𝑥𝑥 4 𝑥𝑥 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 40𝑥𝑥 ! 30𝑥𝑥 135? Select all thatapply.""""! !! !! !!!"""!!!!!!Solve for 𝑦𝑦: 200𝑦𝑦 ! 900𝑦𝑦 1000!Section 6: Quadratics – Part 1133
2. The area of the rectangle is 105 square units.What is the value of 𝑥𝑥?2𝑥𝑥 1𝑥𝑥Section 6 – Video 6Solving Quadratics by Factoring – Special CasesThere are a few special cases when solving quadratics byfactoring.Perfect Square Trinomials! 𝑥𝑥 ! 6𝑥𝑥 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 𝑎𝑎! 2𝑎𝑎𝑎𝑎 𝑏𝑏 ! .134Section 6: Quadratics – Part 1
Factoring a Perfect Square Trinomial! If 𝑎𝑎! 2𝑎𝑎𝑎𝑎 𝑏𝑏 ! is a perfect square trinomial, then𝑎𝑎! 2𝑎𝑎𝑎𝑎 𝑏𝑏 ! 𝑎𝑎 𝑏𝑏 ! .! If 𝑎𝑎! 2𝑎𝑎𝑎𝑎 𝑏𝑏 ! is a perfect square trinomial, then𝑎𝑎! 2𝑎𝑎𝑎𝑎 𝑏𝑏 ! 𝑎𝑎 𝑏𝑏 ! .Let’s Practice!Determine whether the following expressions are perfectsquare trinomials.Solve for 𝑞𝑞: 𝑞𝑞! 10𝑞𝑞 25 0Solve for 𝑤𝑤: 4𝑤𝑤 ! 49 28𝑤𝑤16𝑥𝑥 ! 44𝑥𝑥 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.𝑥𝑥 ! 8𝑥𝑥 64!Section 6: Quadratics – Part 1135
Try It!Let’s Practice!Solve for 𝑥𝑥: 𝑥𝑥 ! 16 0Solve the equation 49𝑘𝑘 ! 64.Difference of SquaresUse the distributive property to multiply the following binomials.Try It!(𝑥𝑥 5)(𝑥𝑥 5)Solve each quadratic equation.0 121𝑝𝑝! 100(2𝑥𝑥 7)(2𝑥𝑥 7)(5𝑥𝑥 1)(5𝑥𝑥 1)𝑝𝑝! 144 0Describe any patterns you notice.! When we have a binomial in the form 𝑎𝑎! 𝑏𝑏 ! , it is calledthe difference of two squares. We can factor this as𝑎𝑎 𝑏𝑏 𝑎𝑎 𝑏𝑏 .136Section 6: Quadratics – Part 1
BEAT THE TEST!1. Which of the following expressions are equivalent to8𝑎𝑎! 98𝑎𝑎? Select all that apply.""""""2(4𝑎𝑎! 49𝑎𝑎)2𝑎𝑎(4𝑎𝑎! 49)2𝑎𝑎(4𝑎𝑎! 49𝑎𝑎)2𝑎𝑎 7 (2𝑎𝑎 7)2 2𝑎𝑎 7 (2𝑎𝑎 7)2𝑎𝑎 2𝑎𝑎 7 (2𝑎𝑎 7)Section 6 – Video 7Solving Quadratics by Taking Square RootsHow would you solve a quadratic equation like the onebelow?2𝑥𝑥 ! 36 0When quadratic equations are in the form 𝑎𝑎𝑎𝑎 ! 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 ℎ 𝑡𝑡 4𝑡𝑡 ! 20𝑡𝑡 25, where 𝑡𝑡 represents time inseconds and ℎ(𝑡𝑡) represents the bird’s height.Part A: At what time will the bird land on the ground?Solve for 𝑥𝑥 by taking the square root.2𝑥𝑥 ! 36 0Part B: What ordered pair represents your solution?!Section 6: Quadratics – Part 1137
Try It!Solve 𝑥𝑥 ! 121 0.BEAT THE TEST!1. What is the smallest solution to the equation2𝑥𝑥 ! 17 179?ABCDSolve 5𝑥𝑥 ! 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 ℎ(𝑡𝑡) 16𝑡𝑡 ! 𝑠𝑠, where ℎ(𝑡𝑡)represents the lifebelt’s height from the sea at any giventime, 𝑡𝑡 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?𝑥𝑥 ! 14𝑥𝑥 What value could be added to the quadratic to make it aperfect square trinomial?𝑥𝑥 ! 6𝑥𝑥 𝑥𝑥 ! 30𝑥𝑥 𝑥𝑥 ! 18𝑥𝑥 𝑥𝑥 ! 22𝑥𝑥 71 𝑥𝑥 ! 8𝑥𝑥 3 𝑥𝑥 ! 4𝑥𝑥 57 𝑥𝑥 ! 10𝑥𝑥 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 𝑎𝑎𝑥𝑥 ! 𝑏𝑏𝑏𝑏 𝑐𝑐 is a perfect square trinomial, thenComplete the square to solve the following equations.𝑎𝑎𝑥𝑥 ! 𝑏𝑏𝑏𝑏 𝑐𝑐 𝑥𝑥 ! 6𝑥𝑥 0𝑎𝑎𝑥𝑥 𝑐𝑐!and 𝑎𝑎𝑥𝑥 ! 𝑏𝑏𝑏𝑏 𝑐𝑐 !𝑎𝑎𝑥𝑥 𝑐𝑐 .𝑥𝑥 ! 16𝑥𝑥 0𝑥𝑥 ! 6𝑥𝑥 0 3𝑥𝑥 ! 6𝑥𝑥 02𝑥𝑥 ! 6𝑥𝑥 6140 5𝑥𝑥 ! 30𝑥𝑥 02𝑥𝑥 ! 4𝑥𝑥 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 2𝑥𝑥 ! 24𝑥𝑥 29 0 by completingthe square. Place the equations in the correct order.1. Write the equation in standard form.2. If 𝑎𝑎 does not equal one, divide every term in the equationby 𝑎𝑎.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) 𝑥𝑥 ! 12𝑥𝑥 36 14.5 36E) 𝑥𝑥 6C) 𝑥𝑥 ! 12𝑥𝑥 14.5G)B) 𝑥𝑥 6 50.5D) 𝑥𝑥 6 50.5! 50.5F) 𝑥𝑥 ! 12𝑥𝑥 14.5 0𝑥𝑥 6! 50.56. Factor the trinomial.7. Take the square root of both sides.8. Solve for 𝑥𝑥.!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, 𝑎𝑎𝑥𝑥 ! 𝑏𝑏𝑏𝑏 𝑐𝑐, 𝑐𝑐 is the𝑦𝑦-intercept.This is the 𝑥𝑥-coordinate of the vertex,𝑥𝑥 !!!!, where values of 𝑎𝑎 and 𝑏𝑏come from the standard form of aquadratic equation. 𝑥𝑥 !!!!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 𝑥𝑥-coordinate from thestep above and evaluate to find 𝑦𝑦. Invertex form, the height is 𝑘𝑘 and thevertex is (ℎ, 𝑘𝑘).The 𝑥𝑥-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 the𝑥𝑥- ‐intercept(s).In function 𝐻𝐻 𝑡𝑡 𝑎𝑎𝑡𝑡 ! 𝑏𝑏𝑏𝑏 𝑐𝑐, ifheight is given, then substitute thevalue for 𝐻𝐻(𝑡𝑡). If time is given, thensubstitute for 𝑡𝑡.
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 𝑡𝑡seconds is given by the equation:ℎ 𝑡𝑡 0.08𝑡𝑡 ! 0.64𝑡𝑡 5.15What would the graph of ℎ(𝑡𝑡) versus 𝑡𝑡 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 ℎ 𝑡𝑡 16𝑡𝑡 ! 80𝑡𝑡, whereℎ(𝑡𝑡) is the height of the ball, in feet, and 𝑡𝑡 is the time inseconds it has been in the air.What would the graph of ℎ(𝑡𝑡) versus 𝑡𝑡 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 “U-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.ℎ 𝑡𝑡 (𝑡𝑡 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: ℎ 𝑡𝑡 0.5𝑎𝑎𝑡𝑡 ! 𝑣𝑣𝑣𝑣 𝑠𝑠, where the gravitationalconstant, 𝑎𝑎, is 32 feet per square second, 𝑣𝑣 is the initialvelocity, 𝑠𝑠 is the initial height, and ℎ(𝑡𝑡) is the height in feetmodeled as a function of time, 𝑡𝑡.Part A: What function describes the height, ℎ, of the bottlerocket after 𝑡𝑡 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, 2015 · 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
QUADRATICS UNIT Solving Quadratic Inequalities and Curve Fitting By graphing the inequality: y x2 – 7x 10, we can begin to look at what shading would look like: Looking at the inequality: y –3x2 – 6x – 7, write the solution: We can also use our knowledge of and and or to solve much faster: x2 12x 39 12 x2 – 24 5x or and
1 Solving Quadratics by Factoring (Day 10 . If necessary, express your answers in simplest radical form. 1. x2 2x 12 2. k(x) 2x2 8x 7 3. The perimeter of a triangle can be represented by the expression 5x2 10x 8. Write a polynomial that represents the measure of the third side. 16
Quadratics Unit Test Review Multiple Choice Identify the choice that best completes the statement or answers the question. _ 1. Identify the vertex of the graph. Tell whether it is a minimum or maximum. a. (0, –1); minimum c. (0, –1); maximum b. (–1, 0); maximum d. (–1, 0); minimum _ 2. Which of the quadratic functions has the .
Common Core Math 2 Unit 1A Modeling with Quadratics 4 Common Core Standards A.SSE.1 Interpret expressions that represent a quantity in terms of its context.« a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity.
Paper Reference(s) 6663/01 . Edexcel GCE . Core Mathematics C1 . Advanced Subsidiary . Quadratics . Calculators may NOT be used for these questions. Information for Candidates . A booklet ‘Mathematical Formulae and Statistical Tables’ might be needed for some questions. The marks for the parts of questions are shown in ro
UNIT P1: PURE MATHEMATICS 1 – QUADRATICS 4 If , the parabola has a maximum value. You feel negative so you got a sad face. 4.1 Characteristics of Quadratic functions The general shape of a parabola is the shape of a pointy letter u _, or a sli
Mar 04, 2019 · Algebra 1 10.1 Worksheet Graphing Quadratics Show all work, when necessary, in the space provided. For question 1 - 6, identify the maximum or minimum point, the axis of symmetry, and the roots (zeros) of the graph of the quadratic function shown, as indicated. Se
quadratic, I started by asking for some examples of a quadratic and then moved on to creating a definition of a quadratic. I reviewed some key terms of quadratics and ways to solve a quadratic eq
Unit 2-1 Factoring and Solving Quadratics Learning Targets: Factoring Quadratic Expressions 1. I can factor using GCF. 2. I can factor by grouping. 3. I can factor when a is one. 4. I can factor when a is not equal to one. 5. I can factor perfect square trinomials. 6. I can factor using difference of squares. Solving Quadratic Equations 7.
Ch 62 Factoring Quadratics, an Introduction To reiterate, we have (2x 3)(x 5) 2x 2 10x 3x 15 product of product of product of product of First terms Outer terms Inner terms Last terms 2x 2 13x 15 The key idea to absorb here is that the 2x 2 in the answer is the
The method described below utilizes two ideas: (1) When the product of two or more numbers is zero, then one of the numbers must be zero. (2) Some quadratics can be factored into the product of two binomials, where coefficients and constants are integers. This procedure is called the Zero Product Method or Solving by Factoring. See the
Graphing-quadratics-worksheet-with-answers. SOLVING QUADRATIC EQUATIONS BY FACTORING WORKSHEET ANSWERS. SOLVING LINEAR. INEQUALITIES WORKSHEET KUTA POLYNOMIAL. Mar 19, 2021 — Worksheet Quadratic Equations solve Quadratic Equations by Peting from worksheet graphing quadratics from standard form answer . Aug 29, 2020 — Graphing quadratic
3 Notes Solving Quadratics with Imaginary Numbers.notebook 1 January 11, 2017 Jan 4 9:06 AM Quadratic Functions MGSE9 12.N.CN.7 Solve quadratic equations with real coefficients that have complex solutions by (but not limited to) square
Green Worksheet - Solve the following quadratics using the quadratic formula and match the question to the answer. Don’t forget to rearrange your equation first. Questions: Answers 1) - 16x 3x2 - 12 0 x 2 2) x2 – 4x –
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Interactions 1 READING Listening &SPEAKING WEEK 4 Date Unit Part 2 Part 2 Part2 Part 3 Part 3 Part 4 Part 4 Part 4 Interactions 1 READING Listening &SPEAKING WEEK 5 Date Unit Quiz 1 Quiz 1 Chapter 3-Intro-Part 1:P.42-45 Chapter 4- Intro. Part 1 Before You Listen Part 1 :P.46-48 Part 2 Part 1 : After You Listen Part 2
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Secret Wall O2 Pit to Q2 X2 To Level 7 (X3) A1 Portal to L10 (A2)  Button Q1 From Pit O1 X3 To Level 7 (X1) 0 Pressure Pad Q2 From Pit O2 X4 To Level 5 (X2) Y Nest In the place where you found a lot of Kenkus (bird creatures) is a place called "Nest." After killing both Kenkus, put all ten Kenku eggs on the floor. The wall will disappear, and .