6-2 (Intro into 6-3)
Finding the vertex of a quadratic function without using
technology
Therefore the axis of symmetry is always given by the xcoordinate of the vertex which is always x=
This means it is also the x-coordinate of the vertex. The
y-coordinate can be found by subbing it into the function.
Ex) Consider the function
you should know now: this is STANDARD FORM
a=
b=
c=
as well as
a) direction of opening:______ why?____________
b) wider or more narrow than y =
?____________
c) the coordinates of the y-intercept?_____________
d) the vertex is a maximum or minimum pt.?_________
e) what is the axis of symmetry? If the function is in
STANDARD FORM , it is always
f) what are the coordinates to the vertex
x=-2
Y=
g) Complete the statement : y has a _______ value of
y = __ and it occurs at x = _______.
H) table of values showing vertex:
I)
Sketch its graph with its line of symmetry stating the
domain and range.
Ex2)
Repeat A to I for y =
A)
B)
C)
E)
F)
G)
H)
I)
Your Turn: Repeat in your notes for:
D)
Determining the axis of symmetry from given the xintercepts
Ex) Given the parabola below determine the equation of
the axis of symmetry.
given x-intercepts the axis of symmetry is always the
average of the x-intercepts x = (-2 + 4)/2 = 2/2 =1
AOS : x = 1 (half-way the x-intercepts)
6-3
The Factored Form of a Quadratic Function (FORM 2)
Ex) Using technology , find the x-intercepts to the
following: (read off the x-intercepts from the graph!)
X intercepts (some have 2)
x=
The easiest way to find them(x-intercepts) is in factored
form: the x-intercepts are always the opposite signs of the
indicated factor unless there is a coefficient before x !!!
for
y = (x-6)(x+5) x intercepts are x = ______ and x =____
y = (2x+3)(x-4) x-intercepts are x = _____ and x = ____
Factored Form of a Quadratic function is y=a(x-r)(x-s)
where x=r and x = s are the zeroes of the function (where
the graph crosses the x-axis----so they are the xintercepts as well)
Ex) Determine the intercepts of the quadratic functions
below with and without technology.
A)
x-int:
y-int:
AOS:
Vertex:
B)
C)
Determining the x-intercepts algebraically by Factoring
2nd Form
Factored Form: y = a(x-r)(x-s)
Standard form y =
1)
2)
3)
4)
AOS (Average of x-intercepts)
Vertex (AOS, sub in x-coordinate)
x-intercepts: set factors equal to 0
y-intercept: sub in x = 0
Review of factoring concepts: M 1201
Factoring is always performed in the following steps
GCF, TRI (2types), DOS
Ex) Review Factor completely by removing the gcf.
The GCF is always removed first!
Ex) Factor completely by using trinomials
Process for (A) is:
1) find two numbers that multiply to give
you 6 and have a sum of the middle coefficient 7
2) they are 6 and 1
)
therefore the x-intercepts to:
AOS = (-1+-6)/2=-3.5
B)
C)
D)
E)
Back to 2201
Ex) Determine the intercepts (x and y), the AOS, vertex
by writing the quadratic functions below in
FACTORED form. They are now in ______ FORM!
E)
Determining the x-intercepts , AOS, Vertex from
Factored form continued
6-3 C--ted
Standard F:
Factored F: y= a(x-r)(x-s)
Trinomials that do not have a GCF or have a coefficient
before the quadratic term other than 1 must use the
process of decomposition to factor the trinomial
(NO GCF)
the graph looks like: (stop_!)
G) SQ (x-int, AOS)
y = 4(x + 6)(x - 5)
y=-(x - 4)(x + 3)
6-3
Determining Equations of quadratic functions given the
zeros of the functions or the x-intercepts (zeroes means
values of x that make y = 0)
Ex) Determine the equation of the following quadratic
functions in the specified form:
A) has zeroes (x-intercepts) at x = -3 and 2 and has a yintercept of 6 in FACTORED and STANDARD form
Picture:
Factored:
f(x) = a(x+3)(x-10) since y-intercept is (0,60) x= 0 y = 60
60 = a(0+3)(0-10)
60=-30a
therefore: y=-2(x+3)(x-8) factored form
multiply out for standard form
note a = -2, b = 14 and c = 60
Finding Equations of parabolas continued
April 16,
2014
Ex) Find the equation of the parabola below in factored
and standard form. DOF \ DOE
Factored Form: (x---intercepts) here x= -2,4
x
y=a(x-r)(x-s)=a(x+2)(x-4)
Standard Form: obtained by
multiplying ff out
Ex) Find the equation of the quadratic function that has
x-intercepts at -3 and 40 and contains the point
(-2,8). (What form ? Your choice!)
The Vertex Form of a Quadratic Function
Introduction to the Vertex form of a quadratic function
Consider the base function y = x2 [y=a(x-h)2+k]
graph f(x+2), f(x) +3, F(x-1)-5
Base Graph
Effect
f(x)=x2
Horizontal
Shifts (x)
New Equation Effect on
f(x)=x2
f(x-4)
f(x+5)
f(x-3)
f(x+2)
Vertical Shifts f(x)+3
(y)
f(x)-4
f(x)+6
f(x)+3
f(x)-2
Combinations
f(x+3)-2
f(x-2)+4
f(x+4)+7
f(x-6)-3
Ex) Given
a) find the coordinates of the vertex
b) the equation of the axis of symmetry
c) the domain and range
d) complete y has a ____ value of y = ___ and it occurs at
x = ______
e) give table of values showing 5 points and sketch its
graph
Ex) Repeat for
a) find the coordinates of the vertex
b) the equation of the axis of symmetry
c) the domain and range
d) complete y has a ____ value of y = ___ and it occurs at
x = ______
e) give table of values showing 5 points and sketch its
graph
Ex) Repeat for
Homework: Same Question:
Determining Equations of Parabolas (Quadratic Functions
in Vertex Form, factored and standard form)
Info given:
Best form to Use
1)Vertex and Point
How to get
a?
sub in point
other than
vertex
2) x-intercepts
y=a(x-r)(x-s) factored
3) factors
Same
same
Ex) Find the equation of the following parabolas in the
indicated form:
a)
the function has factors (x-1) and (x+3) and passes
through the point (0,8) (form to use = )
b) contains the vertex (-1,6) and passes through (-2,0)
(Best form =
)
c)
has vertex at (3,-8) and a y intercept of 1
d) has x-intercepts at x = 4 and x= -2 and contains the
point (2, 8)
Given a picture of the graph
e)
determine an equation for the parabola below in all
three forms
Applications of Quadratic Functions
Ex) A rocket is fired into the air and its path can modeled
by the function h(t) =
where t is time in
seconds and h(t) is the height of the rocket in meters.
Determine the maximum height reached by the rocket
and how long it took to reach this height.
EX)
A soccer ball is kicked from the ground. After 2
s, the ball reaches its maximum height of 20 m. It lands
of the ground at 4s. Determine a quadratic function in
vertex form to describe the ball’s path and use it to find
the height of the ball after 3 s.
Ex)
Ex)
EX)