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Investigating transformations of quadratics

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TRANSFORMATIONS OF QUADRATIC FUNCTIONS
Quadratic functions may be expressed in what is known as the Turning Point form:
𝑦 = π‘Ž(π‘₯ βˆ’ β„Ž)2 + π‘˜
In this activity you will explore the effect that these variables a, h and k have on the graph of a quadratic
function.
Open the GeoGebra link: https://www.geogebra.org/m/mk3yXd7C
Set the initial settings as follows:
a = dilation in y-axis
h = horizontal translation
k = vertical translation
Use the sliders to change the values.
Only change one variable at a time. Leave the other two in these
initial settings.
DILATION AND REFLECTION – The effect of changing β€œa”
a
Equation
Sketch
Shape compared to
π’š = π’™πŸ
Axis of
symmetry
equation
π‘₯=0
Turning point
(x, y)
max / min
(0, 0)
min
1
𝑦 = (π‘₯)2
Original basis of
comparison
2
𝑦 = 2(π‘₯)2
Stretch in y-axis
Makes the graph skinnier
For a given value of x, the
y-value has increased by a
factor of 2
π‘₯=0
(0, 0)
min
Compress in y-axis
Makes the graph fatter
For a given value of x, the
y-value has decreased by a
factor of 0.8
π‘₯=0
(0, 0)
min
Reflection across
y = 0 (x-axis)
π‘₯=0
(0, 0)
max
(x, y)β†’(x, 2y)
0.8
2
𝑦 = 0.8(π‘₯)
(x, y)β†’(x, 0.8y)
-1
2
𝑦 = βˆ’1(π‘₯)
(x, y)β†’(x, -y)
QUESTION 1 (ANSWER IN EXERCISE BOOK)
Use the above information to describe what happens to the graph of 𝑦 = (π‘₯)2 (in terms of axis of symmetry and
turning point) when:
a>1
0<a<1
a<0
TRANSLATION – The effect of changing β€œk”
k
Equation
Sketch
Shape compared to
π’š = π’™πŸ
0
𝑦 = (π‘₯)2
Original basis of
comparison
1
𝑦 = (π‘₯)2 + 1
Vertical translation, 1 unit
up the y-axis
For a given value of x, the yvalue has increase by 1 unit
Axis of
symmetry
equation
π‘₯=0
Turning point
(x, y)
max / min
(0, 0)
min
π‘₯=0
(0, 1)
min
π‘₯=0
(0, 3)
min
(x, y)β†’(x, y+1)
3
𝑦 = (π‘₯)2 + 3
Vertical translation, 1 unit
up the y-axis
For a given value of x, the yvalue has increase by 3
units
(x, y)β†’(x, y+3)
-1
𝑦 = (π‘₯)2 βˆ’ 1
Vertical translation, 1 unit
down the y-axis
For a given value of x, the yvalue has decrease by 1
unit
π‘₯=0
(0, -1)
min
-2
𝑦 = (π‘₯)2 βˆ’ 2
Vertical translation, 2 units
down the y-axis
For a given value of x, the yvalue has decrease by 2
units
π‘₯=0
(0, -2)
min
(x, y)β†’(x, y-1)
(x, y)β†’(x, y-2)
QUESTION 2 (ANSWER IN EXERCISE BOOK)
Use the above information to describe what happens to the graph of 𝑦 = (π‘₯)2 (in terms of axis of symmetry and
turning point) when:
k>0
k<0
TRANSLATION – The effect of changing β€œh”
h
Equation
Sketch
Shape compared to
π’š = π’™πŸ
Axis of
symmetry
equation
π‘₯=0
Turning point
(x, y)
max / min
(0, 0)
min
0
𝑦 = (π‘₯)2
Original basis of
comparison
1
𝑦 = (π‘₯ βˆ’ 1)2
Horizontal translation, 1
unit to the right in the xaxis
π‘₯=1
(1, 0)
min
(x, y)β†’(x+1, y)
3
𝑦 = (π‘₯ βˆ’ 3)2 + 3
Horizontal translation, 3
units to the right in the xaxis
π‘₯=3
(3, 0)
min
-1
𝑦 = (π‘₯ + 1)2 βˆ’ 1
Horizontal translation, 1
unit left in the x-axis
π‘₯ = βˆ’1
(-1, 0)
min
Horizontal translation, 2
units left in the x-axis
π‘₯ = βˆ’2
(-2, 0)
min
(x, y)β†’(x+3, y)
(x, y)β†’(x-1, y)
-2
𝑦 = (π‘₯ + 2)2 βˆ’ 2
(x, y)β†’(x-2, y)
QUESTION 3 (ANSWER IN EXERCISE BOOK)
Use the above information to describe what happens to the graph of 𝑦 = (π‘₯)2 (in terms of axis of symmetry and
turning point) when:
h>0
h<0
SUMMARY
The transformations of a quadratic are easily seen when it is expressed in turning point form 𝑦 = π‘Ž(π‘₯ βˆ’ β„Ž)2 + π‘˜
Where:
a = dilation in y-axis
h = horizontal translation
k = vertical translation
Key features of a graph include:
ο‚· The coordinate of the turning point, (h, k) and whether it is a maximum or a minimum
ο‚· The y-intercept (0, y) – a quadratic will only have one y-intercept
ο‚· The x-intercepts (x, 0) – a quadratic may have 0, 1, or 2 x-intercepts.
ο‚· Axis of symmetry
PRACTICE
Use the GeoGebra to complete the following table. When copying the graph into the grid, use a screen shot and
then adjust the picture size in the PICTURE TOOLS > FORMAT tab to be 2.9 cm wide. Don’t worry about the
height, just set the width and the height will automatically adjust.
>>
Equation
𝑦 = 0.5(π‘₯ βˆ’ 3)2 βˆ’ 2
𝑦 = βˆ’2(π‘₯ + 1.5)2 + 1.2
𝑦 = βˆ’π‘₯ 2 + 3
𝑦=
1 2
π‘₯ +2
2
Sketch
a
h
k
2
-1
1.5
-0.3
0
4
-1
-4
0
Axis of
symmetry
equation
Turning
Point
(x, y)
max/min
x-int
(when
y = 0)
y-int
(when
x = 0)
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