L5–2.4–FamiliesofPolynomialFunctionsLessonMHF4UJensenInthissection,youwilldetermineequationsforafamilyofpolynomialfunctionsfromasetofzeros.Givenadditionalinformation,youwilldetermineanequationforaparticularmemberofthefamily.Part1:Investigation1)a)Howarethegraphsofthefunctionssimilarandhowaretheydifferent?

Same Different• 𝑥-intercepts(zeros)• equationshavesamedegree

• 𝑦-intercepts• stretchorcompressionfactors• vertices

b)Describetherelationshipbetweenthegraphsoffunctionsoftheform𝑦 = 𝑘(𝑥 − 1)(𝑥 + 2),where𝑘 ∈ ℝTheyhavethesame𝑥-intercepts.

2)a)Examinethefollowingfunctions.Howaretheysimilar?Howaretheydifferent?

i) 𝑦 = −2(𝑥 − 1)(𝑥 + 3)(𝑥 − 2)ii) 𝑦 = −(𝑥 − 1)(𝑥 + 3)(𝑥 − 2)iii) 𝑦 = (𝑥 − 1)(𝑥 + 3)(𝑥 − 2)iv) 𝑦 = 2(𝑥 − 1)(𝑥 + 3)(𝑥 − 2)

b)Predicthowthegraphsofthefunctionswillbesimilarandhowtheywillbedifferent.Theywillhavethesame𝑥-interceptsbuttheirshapeanddirectionwillbedifferentduetothesignandvalueoftheleadingcoefficient.c)Usetechnologytohelpyousketchthegraphsofallfourfunctionsonthesamesetofaxes.

Afamilyoffunctionsisasetoffunctionsthathavethesamecharacteristics.Polynomialfunctionswiththesamezerosaresaidtobelongtothesamefamily.Thegraphsofpolynomialfunctionsthatbelongtothesamefamilyhavethesame𝑥-interceptsbuthavedifferent𝑦-intercepts(unless0isoneofthe𝑥-intercepts).Anequationforthefamilyofpolynomialfunctionswithzeros𝑎/, 𝑎1, 𝑎2,…, 𝑎5is:

𝑦 = 𝑘(𝑥 − 𝑎/)(𝑥 − 𝑎1)(𝑥 − 𝑎2)… (𝑥 − 𝑎5),where𝑘 ∈ ℝ, 𝑘 ≠ 0

Part2:RepresentaFamilyofFunctionsAlgebraically1)Thezerosofafamilyofquadraticfunctionsare2and-3.a)Determineanequationforthisfamilyoffunctions.𝑦 = 𝑘(𝑥 − 2)(𝑥 + 3)b)Writeequationsfortwofunctionsthatbelongtothisfamily𝑦 = 8(𝑥 − 2)(𝑥 + 3)𝑦 = −3(𝑥 − 2)(𝑥 + 3)c)Determineanequationforthememberofthefamilythatpassesthroughthepoint(1,4).𝑦 = 𝑘(𝑥 − 2)(𝑥 + 3)4 = 𝑘(1 − 2)(1 + 3)4 = 𝑘(−1)(4)4 = −4𝑘−1 = 𝑘𝑦 = −(𝑥 − 2)(𝑥 + 3)2)Thezerosofafamilyofcubicfunctionsare-2,1,and3.a)Determineanequationforthisfamily.𝑦 = 𝑘(𝑥 + 2)(𝑥 − 1)(𝑥 − 3)b)Determineanequationforthememberofthefamilywhosegraphhasa𝑦-interceptof-15.−15 = 𝑘(0 + 2)(0 − 1)(0 − 3)−15 = 𝑘(2)(−1)(−3)−15 = 6𝑘𝑘 = −2.5

d)SketchagraphofthefunctionNegativeleadingcoefficientandodddegreesoitwillextendfromQ2toQ4Part3:DetermineanEquationforaFunctionFromaGraph3)Determineanequationforthequarticfunctionrepresentedbythisgraph.

Tosketchagraph:

• Ploty-intercept• Plotx-intercepts• Usedegreeandleading

coefficienttodetermineendbehaviour

The𝑥-interceptsare−3,−/1,1,and2

𝑦 = 𝑘(𝑥 + 3)(2𝑥 + 1)(𝑥 − 1)(𝑥 − 2)Thegraphpassesthroughthepoint(-1,-6)−6 = 𝑘(−1 + 3)(2(−1) + 1)(−1 − 1)(−1 − 2)−6 = 𝑘(2)(−1)(−2)(−3)−6 = −12𝑘𝑘 = 0.5𝑦 = 0.5(𝑥 + 3)(2𝑥 + 1)(𝑥 − 1)(𝑥 − 2)