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MCR3U1 U3L9 IT’S ALL IN THE FAMILY (Families of Quadratic Functions) PART A ~ INTRODUCTION Recall that a quadratic function can be expressed in three different algebraic forms: 1. 2. 3. π (π₯) = ππ₯ 2 + ππ₯ + π π (π₯) = π(π₯ − π)(π₯ − π ) π (π₯) = π(π₯ − β)2 + π standard form: factored form: vertex form: family of parabolas: a group of parabolas that all share a common characteristic PART B ~ INVESTIGATE DIFFERENT FAMILIES OF QUADRATIC FUNCTIONS A. Consider the following group of functions: Write each function in factored form: π (π₯) = π₯ 2 − 3π₯ − 10 π (π₯) = π₯ 2 − 3π₯ − 10 π(π₯) = −2π₯ 2 + 6π₯ + 20 β(π₯) = 4π₯ 2 − 12π₯ − 40 π(π₯) = −2π₯ 2 + 6π₯ + 20 β(π₯) = 4π₯ 2 − 12π₯ − 40 1. How are these functions related? ______________________________________________________ 2. What single value was varied to create this family of parabolas? ___________________ B. Consider the following group of functions: Write each function in vertex form: π (π₯) = −2π₯ 2 + 4π₯ + 1 π (π₯) = −2π₯ 2 + 4π₯ + 1 π(π₯) = −6π₯ 2 + 12π₯ − 3 β(π₯) = 10π₯ 2 − 20π₯ + 13 π(π₯) = −6π₯ 2 + 12π₯ − 3 β(π₯) = 10π₯ 2 − 20π₯ + 13 1. How are these functions related? ______________________________________________________ 2. What single value was varied to create this family of parabolas? ___________________ MCR3U1 C. U3L9 Consider the following group of functions: π (π₯) = −3π₯ 2 + 5π₯ − 2 π(π₯) = 2π₯ 2 + π₯ − 2 β(π₯) = 7π₯ 2 − 2π₯ − 2 1. How are these functions related? ______________________________________________________ 2. What values were varied to create this family of parabolas? ________________________ SUMMARY: A. If the value of a is varied in a quadratic function expressed in factored form, π (π₯) = π(π₯ − π)(π₯ − π ), a family of parabolas with the same x–intercepts & axis of symmetry is created. B. If the value of a is varied in a quadratic function expressed in vertex form, π (π₯) = π(π₯ − β)2 + π , a family of parabolas with the same vertex & axis of symmetry is created. C. If the values of a & b are varied in a quadratic function expressed in standard form, π(π₯) = ππ₯ 2 + ππ₯ + π , a family of parabolas with the same y–intercept is created. PART C ~ DETERMINING THE EQUATION OF A QUADRATIC FUNCTION Ex ο Determine the equation of the parabola with: a) x–intercepts 3 and –2 and passing through the point (4, –3) NOTE: There are infinite parabolas with x–intercepts 3 and –2, but only one that will also go through the point (4, –3). MCR3U1 U3L9 b) vertex (–5, –6) and passing through the point (1, –2) To determine the algebraic model of a quadratic function: ο· if the x–intercepts & a point are given, use factored form ο· if the vertex & a point are given, use vertex form Ex ο A highway overpass has a shape that can be modelled by the equation of a parabola. If the highway is 10m wide and the height of the overpass 2m from the edge of the highway is 13m, determine the equation of the parabola modelling the overpass. HOMEWORK: p.192–193 #1–5, 8, 10, 14