Introduction to Linear Graphs
Welcome! In this chapter of Further Pure Mathematics (FPP1), we are going to learn a very clever trick. In the real world, many things don't happen in a straight line—they follow curves, like the path of a ball or the growth of bacteria. However, curves are hard to analyze just by looking at them. Linearization is the process of taking a curved relationship and "stretching" it into a straight line. Why? Because straight lines are easy to measure! By the end of these notes, you’ll be able to turn complex equations into the simple \( y = mx + c \) format to find hidden constants.
1. The Core Idea: Back to Basics
Before we dive into the complex stuff, remember the golden rule of straight lines:
\( Y = mX + c \)
- \( Y \): The value on the vertical axis.
- \( X \): The value on the horizontal axis.
- \( m \): The gradient (slope).
- \( c \): The vertical intercept (where it hits the Y-axis).
In Further Maths, our "vertical axis" might not just be \( y \); it could be \( y^2 \) or \( \log y \). Similarly, our "horizontal axis" could be \( x^3 \) or \( \frac{1}{x} \). Don't worry if this seems tricky at first—it's just like changing the labels on a graph!
2. Reducing Algebraic Relations
Sometimes, we can turn a curve into a line just by moving terms around. Let's look at two common types mentioned in your syllabus.
Type A: The Reciprocal Form
Example: \( \frac{1}{x} + \frac{1}{y} = k \)
To make this look like \( Y = mX + c \), we need to isolate a "Y-variable" on one side.
Subtract \( \frac{1}{x} \) from both sides:
\( \frac{1}{y} = -\frac{1}{x} + k \)
Now, compare it to the straight-line equation:
- Our Vertical Axis (\( Y \)) is \( \frac{1}{y} \)
- Our Horizontal Axis (\( X \)) is \( \frac{1}{x} \)
- Our Gradient (\( m \)) is \( -1 \)
- Our Intercept (\( c \)) is \( k \)
Type B: The Polynomial Form
Example: \( y^2 = ax^3 + b \)
This is already in a linear format if we choose our axes correctly!
Compare \( y^2 = a(x^3) + b \) to \( Y = mX + c \):
- Plot \( y^2 \) on the vertical axis.
- Plot \( x^3 \) on the horizontal axis.
- The gradient will give you the value of \( a \).
- The intercept will give you the value of \( b \).
Quick Review: To linearize, identify which part of the equation can act as your "new" \( Y \) and \( X \). Everything else must be a constant (the gradient or the intercept).
3. Using Logarithms to Linearize
When the variable is an exponent (like \( 2^x \)) or has a power (like \( x^5 \)), we use logarithms to bring the powers down to earth. For this syllabus, we usually use base 10, written as \( \log \).
Memory Aid: The Log Rules
1. \( \log(AB) = \log A + \log B \) (Multiplication becomes Addition)
2. \( \log(A^n) = n \log A \) (Powers move to the front)
Form 1: The Power Law \( y = ax^n \)
This is used when \( x \) is the base and the power \( n \) is a constant (e.g., \( y = 5x^2 \)).
- Take logs of both sides: \( \log y = \log(ax^n) \)
- Split the right side: \( \log y = \log a + \log(x^n) \)
- Move the power: \( \log y = n \log x + \log a \)
Linear Graph: Plot \( \log y \) against \( \log x \).
- Gradient (\( m \)) = \( n \)
- Intercept (\( c \)) = \( \log a \)
Form 2: The Exponential Law \( y = ab^x \)
This is used when the variable \( x \) is the power (e.g., growth of a virus).
- Take logs of both sides: \( \log y = \log(ab^x) \)
- Split the right side: \( \log y = \log a + \log(b^x) \)
- Move the power: \( \log y = (\log b)x + \log a \)
Linear Graph: Plot \( \log y \) against \( x \).
- Gradient (\( m \)) = \( \log b \)
- Intercept (\( c \)) = \( \log a \)
Did you know? This is exactly how scientists determine the half-life of radioactive materials or the rate of interest in a bank account!
4. Step-by-Step: Estimating Constants from Data
If you are given a table of numerical values for \( x \) and \( y \), follow these steps to find the unknown constants:
Step 1: Transform the data. If you've decided to plot \( \log y \) against \( \log x \), use your calculator to find the log of every number in your table.
Step 2: Draw the line of best fit. Plot your new transformed points on a graph. They should form a straight line.
Step 3: Calculate the gradient (\( m \)). Pick two points on the line (not necessarily from your table) and use \( m = \frac{Y_2 - Y_1}{X_2 - X_1} \).
Step 4: Find the intercept (\( c \)). See where the line hits the vertical axis.
Step 5: Solve for constants. If your intercept \( c = \log a \), then find \( a \) by calculating \( 10^c \).
Common Mistake: Don't forget that if your intercept is \( \log a \), the value of \( a \) is not the intercept itself! You must "undo" the log by using the power of 10.
5. Summary and Key Takeaways
Key Points to Remember:
- Linearizing helps us find unknown constants \( a \), \( b \), or \( n \) from curved data.
- Power Law \( y = ax^n \): Plot \( \log y \) vs \( \log x \). Gradient is \( n \).
- Exponential Law \( y = ab^x \): Plot \( \log y \) vs \( x \). Gradient is \( \log b \).
- Always compare your rearranged equation to \( Y = mX + c \) to identify what your axes, gradient, and intercept represent.
Quick Review Box:
If a graph of \( \log y \) vs \( \log x \) has a gradient of 3 and an intercept of 2:
- The power \( n = 3 \).
- The constant \( a = 10^2 = 100 \).
- The original equation was \( y = 100x^3 \).