A' Plotting curvilinear graphs

1
A. Plotting curvilinear graphs

• Plot the graph of the equation y x2 5.

Calculator required
For values of x between 0 and 5.
Set up a table for x and y
1
0
2
3
4
5
x 0 1 2 3 4 5 y
14
21
30
6
9
5
Substitute each value of x in the equation to
find its respective value of y.
y x 2 5
( )
5
6
9
14
21
30
2

x 0 1 2 3 4 5 y
21
30
14
6
9
5
Plot the graph of the equation y x2 5
y
35
x
30
(5, 30)
y x2 5
25
x
20
(4,21)
15
x
(3, 14)
10
x
(2, 9)
x
x
5
(1, 6)
(0, 5)
x
4
5
6
0
1
3
2
0
3
B. Finding a quadratic equation.

• The previous curves drawn are segments of what
  are called
• Parabolas.

Key word
Key word
A parabola is described by plotting a quadratic
equation
y ax2 bx c
4
Worth remembering
y ax2 bx c
Constants or coefficients of x
Key words
The x2 term indicates that a given equation is
quadratic.
Click above for exploremath.com
5
y ax2 bx c
Calculator required

• Find the equation for the following sequence

At this time there is no clue as to whether this
sequence of numbers can be generated by a
quadratic equation or not
1st differences
X y 0 5 1 6 2 9 3 14 4 21 5 30
1
Find the differences between the values of y.
3
5
7
9
6
y ax2 bx c
Calculator required

• Find the equation for the following sequence

1st differences
2nd differences
X y 0 5 1 6 2 9 3 14 4 21 5 30
1
2
If the differences were all the same then the
sequence of y values would be generated by a
linear equation y mx c. Find a second set of
differences.
3
2
5
2
7
2
9
The 2nd differences are constant. The equation
is quadratic.
7
y ax2 bx c
Calculator required

• Find the equation for the following sequence

1st differences
2nd differences
X y 0 5 1 6 2 9 3 14 4 21 5 30
1
a ½ (2nd difference)
2
3
a ½ (2) a 1
2
5
2
7
When x 0, y c
2
9
When x 0, y 5
c 5
8
substitute
Calculator required

• Find the equation for the following sequence

1st differences
2nd differences
X y 0 5 1 6 2 9 3 14 4 21 5 30
y ax2 bx c
a 1 and c 5
1
2
1
6
1
y x2 bx 5.
3
2
( ) ( )2 b( ) 5
5
2
6 1 b 5
7
2
6 6 b
9
b 0
y x2 5.
0 x 0
9
substitute again!
Calculator required

• Find the equation for the following sequence

1st differences
2nd differences
X y 0 5 1 6 2 9 3 14 4 21 5 30
y ax2 bx c
1
y x2 5.
2
1
3
2
9
2
Time for checkin
5
2
Let x 2
7
2
( ) ( )2 5
9
9 4 5
10
C. Trial and improvement.

• The area of the rectangle below is 80 cm2.

Calculator required
x
Let x 5
80 x ( x 2)
X - 2
80 5 ( 5 2)
X is too small
80 15
Let x 12
Find the area by trial and improvement.
80 12 (12 2)
Area of rectangle length width
X is too big
80 120
80 x ( x 2)
Q What is the value of x