Curve Fitting (Grade 12 NSC Matric Mathematics): Revision Notes

Curve Fitting

Introduction to curve fitting

Curve fitting is the process of identifying the mathematical relationship between two variables by examining their scatter plot pattern. When we have bivariate data (data involving two variables), we can plot these as points on a coordinate system to create a scatter plot.

Scatter plots allow us to visualise the direction and strength of relationships between variables:

Types of relationships

When examining scatter plots, we can identify several common relationship patterns that help determine the most appropriate mathematical model for our data:

Linear relationships

• Strong positive linear: Points form a clear upward trend from left to right
• Strong negative linear: Points form a clear downward trend from left to right
• Weak positive linear: Points show an upward trend but with more scatter
• Weak negative linear: Points show a downward trend but with more scatter

Non-linear relationships

• Exponential: Points form a curved pattern that increases rapidly (or decreases rapidly for negative exponential)
• Quadratic: Points form a U-shaped or inverted U-shaped curve (parabola)

No relationship

• Points appear randomly scattered with no recognisable pattern

Intuitive curve fitting

For simple datasets, we can draw a line of best fit (also called a trend line) by hand. This line should pass as close as possible to all data points, with approximately equal numbers of points above and below the line.

Steps for drawing a line of best fit by hand:

1. Plot all data points on a coordinate system
2. Draw a straight line that best represents the overall trend
3. Ensure the line has roughly equal numbers of points above and below it
4. Calculate the equation of the line using two points on the line

The equation of a straight line is: $y = mx + c$

Where:

• m is the gradient (slope)
• c is the y-intercept

To find the gradient: $m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$

Interpolation and extrapolation

Once we have established a relationship between variables, we can make predictions using two distinct approaches:

Interpolation is predicting values that fall within the range of our existing data. This is generally reliable because we're making predictions based on observed patterns.

Extrapolation is predicting values beyond the range of our data. This must be done with caution unless we're confident the relationship continues beyond our data range.

Linear regression

For more precise curve fitting, we use the least squares method. This mathematical technique finds the line that minimises the sum of squared distances between data points and the line.

The least squares regression line has the equation: $y = a + bx$

Where the coefficients are calculated using:

$b = \frac{n\sum xy - \sum x\sum y}{n\sum x^2 - (\sum x)^2}$

$a = \bar{y} - b\bar{x}$

Where:

• n = number of data points
• $\bar{y}$ = mean of y-values
• $\bar{x}$ = mean of x-values

Calculator methods

Scientific calculators can perform linear regression automatically, saving time and reducing calculation errors:

Using SHARP calculator:

1. Change mode to "Stat xy"
2. Enter data pairs using (x,y) and DATA buttons
3. Press RCL then 'a' for y-intercept, RCL then 'b' for gradient

Using CASIO calculator:

1. Press MODE and select STAT
2. Enter data in X and Y columns
3. Access regression menu to get coefficients

Real-world applications

Curve fitting has many practical applications across diverse fields, making it a valuable tool for understanding relationships in real-world scenarios:

• Business growth: Modelling customer acquisition over time (often exponential)
• Medical research: Analysing drug effectiveness vs dosage
• Agriculture: Understanding optimal growing conditions
• Economics: Studying relationships between variables like price and demand