Parabola - Merit
Mahobe
Basics first
y=x
2
y
=
x
+
2
Movement in y direction
2
y-2 = x
2
Movement in x direction
y = ( x + 2)
2
y = x + 4x + 4
2
Reflection in x-axis
y = -x
2
Stretch in y-direction e.g. height doubles
y = 2x
2
1
2
y= x
2
Stretch in x-direction e.g. width halves
y = ( 2x )
2
Sketch
y - 2 = ( x + 4)
2
Sketch
y - 2 = ( x + 4)
2
Sketch
y = - ( x - 3) +1
2
Sketch
y = - ( x - 3) +1
2
Sketch
y = 2 ( x +1) - 4
2
Sketch
y = 2 ( x +1) - 4
2
Factored form of a quadratic
• Draw
y = ( x - 5 ) ( x + 3)
y = ( x - 5 ) ( x + 3)
• Find the intercepts by putting x = 0 and y = 0
• Y-intercept is (0, -15)
• X-intercepts are (5, 0) and (-3, 0)
• The line of symmetry is half way between
these points at x = 1 and y = -16
y = ( x - 5 ) ( x + 3)
• Find the intercepts by
putting x = 0 and y = 0
• Y-intercept is (0, -15)
• X-intercepts are (5, 0) and
(-3, 0)
• The line of symmetry is
half way between these
points at x = 1 and y = -16
Sketch these graphs
y = x ( x - 2)
y = x ( x - 2)
y = 2x - 8
2
y = 2x - 8
2
y = -x ( x -1) - 3
y = -x ( x -1) - 3
• Note that this is just
y = -x ( x -1)
• Moved down 3
y = -x ( x -1) - 3
y = ( x - 2)( x + 4 )
y = ( x - 2)( x + 4 )
y= 2+ x- x
2
y= 2+x-x
2
= - ( x - x - 2)
2
= - ( x - 2 ) ( x + 1)
y = 2 + x - x2
= - ( x2 - x - 2)
= - ( x - 2 ) ( x + 1)
Sketch the following graphs with their axis of
symmetry and give the coordinates of the vertex
y = x - 7x + 6
2
y = x - 7x + 6 = ( x - 6 )( x -1)
2
Vertex (3.5, -6.25)
y = x + 8x - 20
2
y = x + 8x - 20 = ( x +10 ) ( x - 2 )
2
Vertex (-4, -36)
y = x - 2x - 35
2
y = x - 2x - 35 = ( x - 7 ) ( x + 5 )
2
Vertex (1, -36)
y = x - 3x
2
y = x - 3x = x ( x - 3)
2
Vertex (1.5, -2.25)
A is (0, -6) or if the diagram is to scale
(1, -4)
B (-3, 0)
C (2, 0)
D (-0.5, 0)
E (-0.5, -6.25)
A stone is fired from a catapult. The height gained by
the stone is given by the equation
h = 25t - 5t
2
• h= height of the stone
• t = time in seconds
• At what times is the stone at a height of 25
metres?
h = 25t - 5t = 5t ( 5 - t )
2
Use the calculator to solve and round to appropriate level:
h = 25t - 5t = 25
2
t - 5t + 5 = 0
t = 1.4, 3.6
2
What is the stone’s height after 2.5 seconds?
Use the calculator to solve and round to appropriate level:
h = 25t - 5t
2
= 25 ´ 2.5 - 5 ´ 2.5
= 31.25
2
Owen and Becks are playing football. Owen receives a pass and quickly
kicks the ball towards Becks. The graph below shows the path of the
ball as it travels from Owen to Becks. The graph has the equation
y = 0.1( 5 - x ) ( x +1)
Find the value of the y-intercept and explain what this value
represents.
y = 0.1( 5 - x ) ( x +1)
X = 0 y = 0.5 This means the ball’s initial height was 0.5 m
y = 0.1( 5 - x ) ( x +1)
Find the maximum height that the ball reaches.
y = 0.1( 5 - x ) ( x +1)
Halfway between 5 and -1 is 2. Substitute x = 2. the height is 0.9
metres above the ground.
y = 0.1( 5 - x ) ( x +1)
The graphs of y = -x and y = x(x + 2) are shown. Write down the
co-ordinates of A and B.
The graphs of y = -x and y = x(x + 2) are shown. Write down the
co-ordinates of A and B.
A(-3, 3)
B(-2, 0)
Michael throws a cricket ball. The height of the ball follows the
equation: h = 20x – 4x2 where h is the height in metres that the
ball reaches and x is the time in seconds that the ball is in the air.
Describe what happens to the ball: What is the
greatest height? How long is it in the air?
Michael throws a cricket ball. The height of the ball follows the
equation: h = 20x – 4x2 where h is the height in metres that the
ball reaches and x is the time in seconds that the ball is in the air.
h = 4x ( 5 - x )
Maximum height is 25 metres and the ball is in the air
for 5 seconds.
When x = 2, y = 8, so the truck can
travel through the tunnel.
A theme park roller-coaster ride includes a parabolic shaped drop into
a tunnel from a height of 45 metres. This drop can be modelled by
y = x2 – 14x +45. Draw the graph.
y = x 2 -14x + 45
= ( x - 9)( x - 5)
Where does the bottom of the drop occur?
y = x 2 -14x + 45
= ( x - 9)( x - 5)
The bottom of the drop is at 7 metres.
y = x 2 -14x + 45
= ( x - 9)( x - 5)
How many metres does the roller-coaster drop from top to
bottom?
y = x 2 -14x + 45
= ( x - 9)( x - 5)
From 45 to -4. A height of 49 metres.
y = x 2 -14x + 45
= ( x - 9)( x - 5)
Write x2 -14x + 45 in perfect square form.
y = x 2 -14x + 45
= ( x - 9)( x - 5)
Write x2 -14x + 45 in perfect square form.
y = x 2 -14x + 45
= ( x - 7) - 4
2
Find the equation of the following
parabolas.
Don’t forget the stretch
y = -2 ( x - 1) + 18
2
y = -2 ( x - 4 ) ( x + 2 )
y = 2 ( 4 - x )( x + 2 )
1
y = - ( x - 10 ) ( x - 16 )
2
1
2
y = - ( x - 13) + 4.5
2
H = -kx 2 + 3.5
x = -4, H = 1.5
1.5 = -16k + 3.5
1
k=
8
1 2
H = - x + 3.5
8
x = 2, H = 3
Gyn cannot reach the ball as he can
only reach to a height of 2.7 m