Second degree Parabola and more general curve

1. Second Degree Parabola And
More General Curve

2. What is a Curve?
• A curve is a smooth, continuous line with no sharp corners.
Why Study Curves?
• They model real-world phenomena (e.g., bridges, ball
trajectories).
• Help solve problems in physics, engineering, and economics.
• Example:A circle, a straight line, or a wavy road.

3. Second Degree Parabola
• Let (xi , yi ) , i = 1, 2 ,...,n be the
set of n values and let the
relation between x and y be y =
a + bx + cx2
• The constants a, b, and care
selected such that the parabola
is the best fit to the data. The
residual at x = xi is

4. Second Degree Parabola

5. • These equations are known as normal equations. They can be solved
simultaneously to determine the best values of a, b, and c.
• Once these values are obtained, the best-fitting parabola is given by the
equation: y=a+bx+cx2
• Substituting the calculated values of a, b, and c into this equation provides
the required parabolic curve that best fits the given data.

6. Example
• Find the least squares quadratic curve y=a+bx+cx2 for the data:
x 0 1 2 3 4
y 1 1.8 1.3 2.5 6.3
• ∑y = na + b ∑x + c ∑ x ^ 2
• ∑xy = a ∑x + b ∑ x ^ 2 + c ∑x ^ 3
• ∑x ^ 2 y = a x ^ 2 + b ∑x ^ 3 + c ∑x ^ 4
Solution .

7. ANALYSIS
X Y x2
x3
X4
xy
x2
y
0 1 0 0 0 0 0
1 1.8 1 1 1 1.8 1.8
2 1.3 4 8 16 2.6 5.2
3 2.5 9 27 81 7.5 22.5
4 6.3 16 64 256 25.2 100.8
∑x=10 ∑y=12.9 ∑x2
=30 ∑x3
=100 ∑x4
=354 ∑xy=37.1 ∑x2
y=130.3

8. • Normal Equations:
12.9=5a+10b+30c
37.1=10a+30b+100c
130.3=30a+100b+354c
• Equation:
y=1.42−1.07x+0.55x2
• Solution:
a=1.42a=1.42
b=−1.07b=−1.07
c=0.55c=0.55

9. • Let (x, y), i = 1, 2, ..., n be the set of n values and let the relation between x and y be y = abx
.
• Taking logarithm on both the sides of the equation y = abx
.
logey=loge a+ x logeb
Putting loge y = Y, loge a = A, x = X, and loge = B
Y = A + BX
• This is a linear equation in X and Y. The normal equations are
ΣΥ = ∑Α+ΒΣx
ΣΧΥ = ΑΣΧ+ΒΣx2
• Solving these equations, A and B, and, hence, a and b can be found. The best fitting exponential
curve is obtained by substituting the values of a and b in the equation y = abx
.
• Similarly, the best fitting exponential curves for the relation y= axb
and y= aebx
can be obtained.

10. • y = axb
• Taking logarithm on both the sides
logey=loge a+ blogex
• Putting logey=Y, logea=A,b=B and loge x=X ,
Y=A+BX
• The normal equations are
ΣΥ = n A+ΒΣx
ΣΧΥ = ΑΣΧ+ΒΣx2

11. THANK
YOU