# IT’S ALL IN THE FAMILY (Families of Quadratic Functions)

```MCR3U1
U3L9
IT’S ALL IN THE FAMILY
PART A ~ INTRODUCTION
Recall that a quadratic function can be expressed in three different algebraic forms:
1.
2.
3.
π (π₯) = ππ₯ 2 + ππ₯ + 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? ___________________
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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 &amp; 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 &amp; axis of symmetry is created.
C. If the values of a &amp; b are varied in a quadratic function expressed
in standard form, π(π₯) = ππ₯ 2 + ππ₯ + 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).
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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 &amp; a point are given, use factored form
ο· if the vertex &amp; 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
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