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The vertex form of a quadratic function
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(Vertex, range, axis of symmetry, max and min
value, and word problems) |
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Team members |
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Name:
Ali Haider
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Overview |
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This section is all about
quadratic function. A quadratic function
is a real or complex function f with
the functional equation
f(x)=ax2+bx+c,
where a≠0,
b, and c are real (or complex)
numbers. The quadratic function also can be
written in the vertex form as
f(x)=a(x-p)2+q
, where (p , q)
is the vertex. The graph of a quadratic function
is called a parabola.
In terms of standard type
of questions, by the end of this section the
student should be able to determine the maximum
and minimum value of a quadratic function. Also,
they should know how to find the vertex, axis of
symmetry, range, domain, direction of opening,
and y and x intercepts. In addition, the
students should learn how solve world problems
related to quadratic functions, because there
will be lots of word problems in this section.
In order to solve these kinds of questions the
students must learn how to change equations from
standard form to vertex form.
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Study Tips,
Methods and or Advice |
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The most important thing in
this section is learning how to write a
quadratic function in its vertex form, because
the vertex is a very important point which can
help us a lot in this section. By finding the
vertex, we will be able to determine the
range, maximum and minimum value
and also axis of symmetry.
Axis of symmetry is a line
that divides our parabola in to two congruent
parts, and the vertex is the point where the
axis of symmetry of a parabola intersects the
parabola.
To determine the vertex of
the graph of a quadratic function,
f(x) = ax2+
bx + c first we have to write it
in the vertex form
f(x) = a(x-p)2+q
by
completing the square. Therefore
the vertex will be
(p , q)
.
In vertex, the value of
p
will determine the axis of symmetry, so it will
be written as
x=p. also
q
represent two important things. It can show us
the range and Max or Min
value of our graph.
When
a > 0,
the parabola opens up and the function has a
minimum value of
q,
and also the range will be written as.
When
a < 0,
the parabola opens down and the function has a
maximum value of
q, and
the range will be written as.
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Sample
Questions and complete solutions |
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Questions |
Complete Solutions |
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1. For
the function
state
its maximum or minimum value and
identify which it is.
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This
function is already in the vertex form.
The
vertex is ( 1 , 2 ).
From the
negative sign in front of a
which is 0.5 we can see
that the graph opens down, therefore we
will have a maximum value
of 2. |
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2.
Without sketching the graph, determine
the maximum or minimum value of the
function
,
state which it is, and state the
corresponding x-value.
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Since
the coefficient of x2
is negative, the parabola opens
down and the function has a maximum
value.
First we
have to write the function in its vertex
form by completing the square.
Therefore the vertex is (-4 , 14)
Axis of
symmetryà
x= - 4 Max
valueà
y=14 |
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3. A
ball is thrown into the air from a
building and falls to the ground below.
The height of the ball, h
meters, relative to the ground t
seconds after being thrown is given by
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a)
Determine the maximum height of the ball
above the ground.
b)
How long does it take the ball to reach
the maximum height?
c)
After how many seconds does the ball hit
the ground?
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Since
the coefficient of x2
is negative, the parabola opens
down and the function has a maximum
value.
First we
have to write the function in its vertex
form by completing the square.
Therefore the vertex is (3 , 80)
a) The
maximum height of the ball above the
ground is 80 meters.
b) It
will take the ball 3 seconds
to reach the maximum height.
c) Solve
for t when h
is 0
Therefore after 7 seconds
the ball will hit the ground. |
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4. Given
the function,
Find:
a)
Vertex
b)
Range
c)
Maximum or minimum value
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First
write the function in its vertex form by
completing the square.
a) The vertex is (2 , -1)
b) The range is
c) Since the graph opens down we have a
maximum value of y= -1
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Extra
Practice Questions and Answers |
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Extra Practice Questions |
Answers |
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1. For the function
state
its maximum or minimum value and
identify which it is.
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-5; maximum
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2. Given the
function,
Find:
a) Vertex
b) Axis of symmetry
c) Direction of opening
d) Range
e) Maximum or minimum value |
a) The vertex is (
3 , 2)
b) x= 3
c) down
d)
e) Maximum; y= 2 |
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3. For the
quadratic function,
i) Write it in the vertex from
ii) Identify the coordinates of the
vertex
iii) Identify the axis of symmetry
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i)
ii) (-12 , 60)
iii) x= -12 |
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4. A rock is thrown into the air from a
bridge and falls to the water below. The
height of the ball, h
meters, relative to the water t
seconds after being thrown is given by
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a)
Determine the
maximum height of the rock above the
water.
b)
How long does it
take the rock to reach the maximum
height?
c)
After how many
seconds does the rock hit the water?
d)
How high is the
bridge above the water?
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a) 35 meters
b) 2 seconds
c) 4.65 seconds
d) 15 meters |
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5. Find the vertex point, axis of
symmetry, x and y intercepts and range
of the given quadratic function:
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Vertex point: (-1 ,
-16)
Axis of symmetry:
x= -1
y-intercept: (0 ,
-15)
x-intercepts:
(-5, 0) and ( 3, 0)
Range:
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6. Find the vertex of the graph of
f(x) = (x + 9)(x - 5)
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(-2, -49) |
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Self
Reflection |
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This section is not hard
section; it is all a review from grade 10, and I
am sure that you will like this section. It will
be very easy for you if you know how to change a
function from the standard form
f(x)=ax2+bx+c
to the vertex form
f(x) = a(x-p)2+q,
and also from vertex to standard. One thing is
very important in this section, and that called
the vertex ( h , k
). If you know how to get this
everything else will be easy. Overall if you do
your home work and pay attention to the teacher
in class you will be okay.
Personally I didn’t have
any problems with this chapter, it was very
easy, and I liked it very much. Overall I liked
this course, it was wonderful. It is not that
much hard if you study and remember doing your
homework on time.
J
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