Quadratic Symmetry
Key words: coefficient, constant, line of symmetry, minimum,
quadratic function,
Objectives
KS4
• Solve word problems and investigate in a variety of
KS3 Framework
contexts and evaluate solutions across a range of contexts.
page ref: 13, 163,
• Plot graphs of simple quadratic functions.
171
• Know simple properties of quadratic functions.
Mathematical skills needed by pupils: Some experience with quadratic functions.
Instructions for teacher:
At the start of the lesson ask students to reset the TI-83 Plus as
follows.
• Press 2nd + to get Mem
• Choose 7 ( Reset)
• Choose 1 (All Ram)
• Choose 2 (Reset)
Create a graphing window with equally spaced x and y
intervals. Press WINDOW and enter the values shown on the
right. Use (─) to enter negative numbers.
This activity asks students to explore the relationship between
the quadratic function, the equation of its line of symmetry and
the coordinates of its lowest point (minimum)
Press Y=.
Enter the function y = x2 – 4x + 3
by pressing X,T,θ,n x2 – 4 X,T,θ,n + 3
Press TRACE.
Use ◄ or ► to move the cursor to the lowest point of the
graph. The coordinates are displayed at the bottom of the
screen.
What is the equation of the line of symmetry? x = 2
Students may need reminding about the equations of vertical
lines.
What are the coordinates of the minimum point?: (2, ─1)
An alternative method for finding the coordinates of the
minimum point is as follows:
Press Y= and enter a function.
Press 2nd TRACE to get CALC, then 3 to choose minimum
The calculator will prompt you: Left Bound?
Move the cursor to the left-hand side of the lowest point and
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press ENTER
The calculator will prompt you: Right Bound?
Move to the right-hand side of the lowest point and press
ENTER
The calculator will prompt you: Guess?
Move close to the lowest point and press ENTER
Answers to student activity sheet
1. y = x2 + 4x + 1
x = ─2
(─2, ─3)
2. y = x2 – 6x + 7
x=3
(3, ─ 2)
3. y = x2 + 10x + 21 x = ─5
(─5, ─4)
4. y = x2 – 10x + 19 x = 5
(5, ─6)
5. y = x2 –2x – 4
x=1
(1, ─5)
6. y = x2 – 8x + 14
x=4
(4, ─2)
7. y = x2 + 2x – 1
x = ─1
(─1, ─2)
8. y = x2 + 12x + 35 x = ─6
(─6, ─1)
9. y = x2 – 14x + 44 x = 7
(7, ─5)
10. y = x2 – 4x + 6
x=2
(2, 2)
11. y = x2 – 5x + 5
x = 2.5
(2.5, ─1.25)
12. y = x2 + 3x – 1
x = ─1.5
(─1.5, ─3.25)
For the function y = x2 + bx + c,
the line of symmetry is x = ─ (b/2)
and the coordinates of the minimum point (─(b/2), c – (b/2)2)
The rule works for both even and odd coefficients of x.
If students have difficulty finding the relationship for the y-coordinate, hint that they
need to square the x-coordinate.
This activity is a useful foundation for work in Year 10 on completing the square.
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Quadratic Symmetry
Make sure that you have a friendly graph screen by pressing WINDOW and
entering the values shown here.
Use (—) to enter a negative number
This screen shows the graph of y = x2 – 4x + 3
The graph is symmetrical.
What is the equation of the line of symmetry?
What are the coordinates of the lowest point of the graph?
Press Y=.
Enter the function y = x – 4x + 3 by pressing X,T,θ,n x2 – 4 X,T,θ,n + 3
2
Press TRACE.
Move the cursor, using ◄ or ►, to the lowest point on the graph.
The coordinates are given at the bottom of the screen
The line of symmetry is x = 2
The lowest point is at (2, —1)
Press Y=, then CLEAR.
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Draw graphs of the functions in the table on your TI-83 Plus.
Record the equation of the line of symmetry
and the coordinates of the lowest point.
Function
Line of symmetry
Lowest point
2
1 y = x + 4x + 1
2 y = x2 – 6x + 7
3 y = x2 + 10x + 21
4 y = x2 - 10x + 19
5 y = x2 – 2x - 4
6 y = x2 – 8x + 14
7 y = x2 + 2x – 1
8 y = x2 + 12x + 35
9 y = x2 – 14x + 44
10 y = x2 – 4x + 6
Look back at your results.
Can you find a way to predict the line of symmetry from the function?
What about finding the lowest point from the function?
So far, all the functions have had an even coefficient of x.
Do your rules work for functions with an odd x-coefficient?
Check with y = x2 – 5x + 5 and y = x2 + 3x – 1
Test these TI-83 Plus commands.
Press Y= and enter any one of the functions from the table.
Press 2nd TRACE to get CALC, then 3 to choose minimum
The calculator will ask you for the: Left Bound?
Move the cursor to the left-hand side of the lowest point and press ENTER
The calculator will ask you for the: Right Bound?
Move to the right-hand side of the lowest point and press ENTER
The calculator will ask you for a; Guess?
Move close to the lowest point and press ENTER
What is the TI-83 Plus finding for you?
Try some of your own functions.
Predict the line of symmetry and the lowest point.
Check by using 2nd TRACE 3
Reset your TI-83 Plus
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