6.1 Linear and Non-Linear Relations
1. Line of Best Fit
When non-linear data is plotted directly, it forms a curve. To analyze non-linear relationships, data is reduced to a linear form $Y = mX + c$ and plotted to obtain a line of best fit.
Characteristics of a line of best fit:
- The line passes through as many points as possible.
- The number of points lying above and below the line should be roughly equal and balanced in distance.
- Extrapolation and interpolation can be performed using the line of best fit to estimate missing values.
6.2 Linear Law and Non-Linear Relations
1. Converting Non-Linear Equations to Linear Form
A non-linear equation relating $x$ and $y$ can be converted to the standard linear form:
$$Y = mX + c$$
where:
- $Y$: Capital variable representing the vertical axis function (e.g., $y$, $\frac{y}{x}$, $xy$, $\lg y$, $\frac{1}{y}$)
- $X$: Capital variable representing the horizontal axis function (e.g., $x$, $x^2$, $\frac{1}{x}$, $\sqrt{x}$, $\lg x$)
- $m$: Gradient of the straight line
- $c$: Vertical $Y$-intercept
2. Common Non-Linear Reductions
Type A: Polynomial / Fractional Forms
- Equation: $y = ax^2 + bx$ $\rightarrow$ Divide by $x$:
$$\frac{y}{x} = ax + b \quad \left(Y = \frac{y}{x}, \, X = x, \, m = a, \, c = b\right)$$
- Equation: $y = \frac{a}{x} + bx$ $\rightarrow$ Multiply by $x$:
$$xy = b x^2 + a \quad \left(Y = xy, \, X = x^2, \, m = b, \, c = a\right)$$
- Equation: $y\sqrt{x} = a + b\sqrt{x}$ $\rightarrow$ Divide by $\sqrt{x}$:
$$y = \frac{a}{\sqrt{x}} + b \quad \left(Y = y, \, X = \frac{1}{\sqrt{x}}, \, m = a, \, c = b\right)$$
- Equation: $\frac{1}{y} = a x^2 + b$
$$\left(Y = \frac{1}{y}, \, X = x^2, \, m = a, \, c = b\right)$$
Type B: Exponential / Power Forms (Using Logarithms)
- Equation: $y = a b^x$ $\rightarrow$ Apply $\log_{10}$ ($\lg$) on both sides:
$$\lg y = \lg(a b^x) = \lg a + \lg(b^x)$$
$$\lg y = (\lg b) x + \lg a \quad \left(Y = \lg y, \, X = x, \, m = \lg b, \, c = \lg a\right)$$
- Equation: $y = a x^b$ $\rightarrow$ Apply $\log_{10}$ ($\lg$) on both sides:
$$\lg y = \lg(a x^b) = \lg a + b \lg x$$
$$\lg y = b(\lg x) + \lg a \quad \left(Y = \lg y, \, X = \lg x, \, m = b, \, c = \lg a\right)$$
- Equation: $p^{x} y = q$ $\rightarrow$ Apply $\log_{10}$ on both sides:
$$\lg y = (-\lg p) x + \lg q \quad \left(Y = \lg y, \, X = x, \, m = -\lg p, \, c = \lg q\right)$$
6.3 Applications of Linear Law
1. Determining Constants from Straight Line Graphs
- Find the gradient $m$ using two points $(X_1, Y_1)$ and $(X_2, Y_2)$ on the line of best fit:
$$m = \frac{Y_2 - Y_1}{X_2 - X_1}$$
- Identify the $Y$-intercept $c$ from the graph or calculate it using $c = Y - mX$.
- Equate $m$ and $c$ to the corresponding terms in the linear transformation to solve for unknown constants (e.g., $a, b, p, q$).