© Teachers Teaching with Technology (Scotland)
Teachers Teaching with Technology
T3 Scotland
Quadratic Functions
Completing the Square
©Teachers Teaching with Technology (Scotland)
QUADRATIC FUNCTIONS OF THE FORM
For each function:
i)
ii)
iii)
iv)
v)
1.
y = a(x + p) 2 + q, when a = 1
Graph on your TI-83.
Sketch the graph on the grid provided.
State whether a Maximum or a Minimum turning point exists
State the coordinates of the turning point.
Give the equation of the Axis of Symmetry
y = (x + 2)2 - 1
Max or Min T.P
Coordinates of T.P
Equation of Axis
of Symmetry
2.
y = (x - 3)2 + 2
Max or Min T.P
Coordinates of T.P
Equation of Axis
of Symmetry
3.
y = -(x + 1)2 - 3
Max or Min T.P
Coordinates of T.P
Equation of Axis
of Symmetry
4.
y = -(x + 2)2 - 1
Max or Min T.P
Coordinates of T.P
Equation of Axis
of Symmetry
T3 Scotland
Quadratic Function: Completing the Square.
Page 1 of 9
PUT DOWN YOUR CALCULATOR
DO NOT GRAPH THIS.
Make a conjecture about the nature of this function.
5.
y = (x + 2)2 - 1
Max or Min T.P
Coordinates of T.P
Equation of Axis
of Symmetry
Now test this conjecture using the Graphic Calculator
PUT DOWN YOUR CALCULATOR
DO NOT GRAPH THIS.
Make a generalisation.
6.
y = (x + p)2 + q
Max or Min T.P
Coordinates of T.P
Equation of Axis
of Symmetry
7.
In your own words describe the effect that the constants p and q have on the graph,
and on when a function of this form has a Maximum or Minimum Turning Point.
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T3 Scotland
Quadratic Function: Completing the Square.
Page 2 of 9
QUADRATIC FUNCTIONS OF THE FORM
For each function:
i)
ii)
iii)
iv)
v)
8.
y = a(x + p) 2 + q, when a ≠ 1
Graph on your TI-83.
Sketch the graph on the grid provided.
State whether a Maximum or a Minimum turning point exists
State the coordinates of the turning point.
Give the equation of the Axis of Symmetry
y = 2(x + 2)2 - 1
Max or Min T.P
Coordinates of T.P
Equation of Axis
of Symmetry
9.
y = 3(x - 3)2 + 2
Max or Min T.P
Coordinates of T.P
Equation of Axis
of Symmetry
10.
y = -2(x + 2)2 - 2
Max or Min T.P
Coordinates of T.P
Equation of Axis
of Symmetry
11.
y = -½(x + 1)2 + 3
Max or Min T.P
Coordinates of T.P
Equation of Axis
of Symmetry
T3 Scotland
Quadratic Function: Completing the Square.
Page 3 of 9
PUT DOWN YOUR CALCULATOR
DO NOT GRAPH THIS.
Make a conjecture about the nature of this function.
12.
y =-2(x + 2) 2 - 1
Max or Min T.P
Coordinates of T.P
Equation of Axis
of Symmetry
Now test this conjecture using the Graphic Calculator
PUT DOWN YOUR CALCULATOR
DO NOT GRAPH THIS.
Make a generalisation.
13.
y = a(x + p)2 + q
Max or Min T.P
Coordinates of T.P
Equation of Axis
of Symmetry
14.
In your own words describe the effect that the constants a ,p and q have on the
graph and when a function has a Maximum or Minimum Turning Point.
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T3 Scotland
Quadratic Function: Completing the Square.
Page 4 of 9
QUADRATIC FUNCTIONS OF THE FORM
y = ax 2 + bx + c, when a = 1
For each function:
i)
Graph on your TI-83.
ii)
Sketch the graph on the grid provided.
iii)
State whether a Maximum or a Minimum turning point exists
iv)
State the coordinates of the turning point.
v)
Give the equation of the Axis of Symmetry
vi)
State the function in the form y = a(x + p)2 + q
15.
y = x2 + 6x + 7
Max or Min T.P
Coordinates of T.P
Equation of Axis
of Symmetry
Function in form
16.
y = a(x + p)2 + q
y = x2 + 2x - 2
Max or Min T.P
Coordinates of T.P
Equation of Axis
of Symmetry
y = a(x + p)2 + q
17.
Function in form
y = -x2 - 4x - 5
Max or Min T.P
Coordinates of T.P
Equation of Axis
of Symmetry
Function in form
18.
y = a(x + p)2 + q
y = x2 - 6x + 11
Max or Min T.P
Coordinates of T.P
Equation of Axis
of Symmetry
Function in form
T3 Scotland
y = a(x + p)2 + q
Quadratic Function: Completing the Square.
Page 5 of 9
QUADRATIC FUNCTIONS OF THE FORM
19.
y = 2x2 - 12x + 20
Max or Min T.P
Coordinates of T.P
Equation of Axis
of Symmetry
y = a(x + p)2 + q
20.
Function in form
y = 3x2 + 6x + 1
Max or Min T.P
Coordinates of T.P
Equation of Axis
of Symmetry
Function in form
21.
y = a(x + p)2 + q
y = 16x - 15 - 3x2
Max or Min T.P
Coordinates of T.P
Equation of Axis
of Symmetry
y = a(x + p)2 + q
22.
Function in form
y = ½x2 - 4x + 10
Max or Min T.P
Coordinates of T.P
Equation of Axis
of Symmetry
Function in form
23.
y = ax2 + bx + c, when a ≠ 1
y = a(x + p)2 + q
In your own words describe the effect that the constants a has on the graph.
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T3 Scotland
Quadratic Function: Completing the Square.
Page 6 of 9
Using your answers from pages 5 and 6.
Copy and complete this table.
a
15.
y = x2 + 6x + 7
16.
y = x2 + 2x - 2
17.
y = -x2 - 4x - 5
18.
y = x2 - 6x + 11
19.
y = 2x2 - 12x + 20
20.
y = 3x2 + 6x + 1
21.
y = 16x - 15 - 3x2
22.
y = ½x2 - 4x + 10
b
c
p
q
2ab
ab2 +c
There are relationships between a,b,c and a,p,q.
State the relationships.
a = _______
p = __________
By stating that the:
General Form of a Quadratic
Completed Square form
and
ax 2 + bx + c
a( x + p)2 + q
q = ____________
equals the
Prove the above relationships ALGEBRAICLY
ax2 + bx + c = a(x + p) 2 + q
=
=
=
T3 Scotland
Quadratic Function: Completing the Square.
Page 7 of 9
Example 1
Complete the square and hence find the turning point and axis of symmetry of the function
y = 2x2 - 8x + 9
The Completed Square form of a Quadratic Equation is:
y = a(x + p)2 + q
p gives the shift of the parabola left or
q gives the shift of the parabola up or
right.
negative is right
positive is left
down.
negative is down
positive is up
y
y = 2x2 - 8x + 9
= 2(x - 2)2 + 1
=2x 2
From the graphic calculator we can see that the graph of y =2x2 has been shifted:
i)
2 to the right, hence p = -2
and
ii)
1 up,
hence q = 1
ALGEBRAIC WORKED SOLUTION
Express the function y = 2x 2 - 8x + 9
in the form
y = a(x + p)2 + q
2 x 2 - 8 x + 9 = a ( x + p) 2 + q
= a ( x 2 + 2 px + p 2 ) + q
= ax 2 + 2apx + ap 2 + q
Comparing coefficients
a =2
2ap = - 8
ap 2 + q = 9
2( 2) p = - 8
( 2 )(-2 )2 + q = 9
4p = - 8
8 +q =9
p=-2
q =1
Therefore
2 x 2 - 8 x + 9 = 2( x - 2) 2 + 1
T3 Scotland
Quadratic Function: Completing the Square.
Page 8 of 9
Completing the Square
Calculator skills sheet
Using a TI - 83 to assist you in completing the square not only reduces the chances of you
making a silly error but also makes the whole process much faster.
Here is a typical question from a textbook and a method for solution.
Find the equation of the axis of symmetry and the coordinates of the turning point of
y = 2 x 2 − 8x + 9
Enter the function on the [Y=] screen and
graph the result on the [ZOOM 4:ZDecimal]
window range.
For some functions [ZOOM 6:ZStandard] is
appropriate.
We can see that the function is a parabola with a
minimum turning point.
We can find the coordinates of the Turning Point
by using the
[2nd][CALC] 3:minimum
or
[2nd][CALC] 4:maximum
facility on the TI-83.
The TI-83 prompts you for a “Left Bound”, a “Right Bound” and a “Guess”.
Using the cursor keys and [ENTER] isolate a section of the graph and give your best guess.
The process the machine carries out is numerical and this information allows it to focus on
value close to the turning point, it is best to make these as accurate as possible.
The coordinates of the minimum turning point are (2,1).
Since the Axis of Symmetry goes throught the turning point it must be the line x =2
T3 Scotland
Quadratic Function: Completing the Square.
Page 9 of 9