Study Notes: Coordinate Geometry (Chapter 4)
Topic C4.2 / E4.2: Gradient of Linear Graphs
Hello! Welcome to the exciting world of Coordinate Geometry. This section is all about understanding the steepness of straight lines—a crucial concept that pops up everywhere, from calculating the pitch of a roof to predicting how fast a rocket is climbing!
Don't worry if this seems tricky at first; the gradient is simply a measure of slope, and we have two easy ways to calculate it.
1. Understanding the Concept of Gradient (Slope)
The gradient (often represented by the letter m) tells us two things about a straight line:
- Steepness: How quickly the line rises or falls. A large gradient (e.g., m = 10) means a very steep line, like a cliff face. A small gradient (e.g., m = 0.5) means a gentle slope, like a ramp.
- Direction: Whether the line goes up or down as you move from left to right.
Analogy: The Hill Climb
Imagine you are walking along a straight path (your line) from left to right:
- If the path goes up, the gradient is Positive (\(m > 0\)).
- If the path goes down, the gradient is Negative (\(m < 0\)).
- If the path is perfectly flat, the gradient is Zero (\(m = 0\)).
Key Definition: Rise and Run
Mathematically, the gradient is defined by the ratio of how much the line moves vertically (the Rise) compared to how much it moves horizontally (the Run).
\[ \text{Gradient} \ (m) = \frac{\text{Rise}}{\text{Run}} = \frac{\text{Vertical Change}}{\text{Horizontal Change}} \]
Key Takeaway
The gradient, m, measures the steepness and direction of a line using the ratio of vertical change (Rise) over horizontal change (Run).
2. Finding Gradient from a Grid (Core Method C4.2)
For Core students, the syllabus requires you to find the gradient directly from a grid by counting squares. This method is based purely on counting the 'Rise' and the 'Run' between two points on the line.
Step-by-Step Process (Counting Squares)
To find the gradient of a line on a graph:
- Choose Two Clear Points: Select two points (\(P_1\) and \(P_2\)) on the line where the coordinates are easy to read (usually where the line crosses grid intersections).
- Determine the Run (Horizontal Change): Count the number of units you move horizontally (left or right) to get from \(P_1\) to \(P_2\). This is the denominator.
- Determine the Rise (Vertical Change): Count the number of units you move vertically (up or down) to get from the end of the Run to \(P_2\). This is the numerator.
- Calculate the Gradient: Divide the Rise by the Run, simplifying the fraction if possible.
Important Note on Signs:
- If you move UP, the Rise is positive (+).
- If you move DOWN, the Rise is negative (-).
- The Run is usually counted moving to the RIGHT, making it positive (+).
Example (Visual):
If a line rises 4 units for every 2 units it runs to the right:
Rise = +4
Run = +2
\[ m = \frac{4}{2} = 2 \]
Quick Review: Special Gradients
- Horizontal Lines: Perfectly flat. The rise is always 0.
Gradient: \(m = \frac{0}{\text{Run}} = 0\). (e.g., \(y = 5\)) - Vertical Lines: Straight up and down. The run is always 0.
Gradient: \(m = \frac{\text{Rise}}{0}\). Since division by zero is impossible, the gradient is Undefined. (e.g., \(x = -3\))
Key Takeaway
When finding the gradient from a graph, always count Rise (vertical change) over Run (horizontal change). Remember to check the sign!
3. Calculating Gradient from Two Coordinates (Extended Method E4.2)
For Extended students (and highly recommended for Core students moving forward), you must know how to calculate the gradient using a formula, even if no grid is provided.
The Gradient Formula
If you have two points, \(P_1 = (x_1, y_1)\) and \(P_2 = (x_2, y_2)\), the gradient m is calculated by the difference in the y-coordinates divided by the difference in the x-coordinates.
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Step-by-Step Process (Using the Formula)
Example: Find the gradient of the line passing through \((3, 10)\) and \((7, 2)\).
- Label Your Points: Assign which point is \(P_1\) and which is \(P_2\). (It doesn't matter which way you choose, as long as you are consistent!)
- Let \(P_1 = (x_1, y_1) = (3, 10)\)
- Let \(P_2 = (x_2, y_2) = (7, 2)\)
- Calculate the Change in Y (\(y_2 - y_1\)):
\(y_2 - y_1 = 2 - 10 = -8\) (The Rise is -8, meaning the line goes down.) - Calculate the Change in X (\(x_2 - x_1\)):
\(x_2 - x_1 = 7 - 3 = 4\) (The Run is 4.) - Divide Rise by Run:
\[ m = \frac{-8}{4} = -2 \]
The gradient of the line is -2. This is a negative gradient, meaning the line slopes downwards.
Memory Aid: The M-Factor
How do you remember the formula? Think of M for Movement (vertical movement first!) or M for Mountain. You always go UP or DOWN the mountain (Y-axis) before you move ACROSS the land (X-axis).
\[ m = \frac{\text{Y-Change}}{\text{X-Change}} \]
Common Mistake to Avoid!
The most frequent error is mixing up the coordinates! Once you label a point as \((x_1, y_1)\), you must start both the numerator and the denominator calculation with the coordinates from the other point, \((x_2, y_2)\).
Example of a mistake: \(m = \frac{y_2 - y_1}{x_1 - x_2}\). DO NOT DO THIS!
Did You Know?
The gradient of a line is exactly the same no matter which two points you choose on that line. This is the definition of a linear graph!
4. Connecting Gradient to the Equation of a Line (C4.4 / E4.4)
The straight-line graph equation is most commonly written in the gradient-intercept form:
\[ y = mx + c \]
- \(m\) is the gradient (steepness).
- \(c\) is the y-intercept (where the line crosses the y-axis).
If you are given the equation in this form, you can immediately state the gradient.
Example:
Find the gradient of the line \(y = 6x + 3\).
Solution: Comparing \(y = 6x + 3\) to \(y = mx + c\), we see that the coefficient of \(x\) is \(m\).
The gradient is \(m = 6\).
What if the equation is not in \(y = mx + c\) form?
Sometimes the equation is given in the form \(ax + by = c\). You need to rearrange it to find the gradient.
Example (Extended Syllabus E4.4): Find the gradient of \(5x + 4y = 8\).
- Isolate the \(y\) term:
\(4y = -5x + 8\) - Divide by the coefficient of \(y\):
\[ y = \frac{-5}{4}x + \frac{8}{4} \]
\[ y = -\frac{5}{4}x + 2 \]
- Identify the gradient:
The gradient \(m\) is the coefficient of \(x\).
\(m = -\frac{5}{4}\) (or -1.25).
Key Takeaway
To easily find the gradient from an equation, always rearrange it into the form \(y = mx + c\). The value of \(m\) is the gradient.
5. Parallel Lines (C4.5 / E4.5)
The concept of parallel lines is directly linked to the gradient.
Parallel lines are lines that travel in exactly the same direction and never meet. This means they have the exact same steepness.
Rule for Parallel Lines:
If Line A is parallel to Line B, then the gradient of A is equal to the gradient of B.
\[ m_A = m_B \]
Example:
Find the equation of the line parallel to \(y = 4x - 1\) that passes through the point \((1, -3)\).
- Find the gradient: The given line \(y = 4x - 1\) has a gradient \(m = 4\).
- Use the parallel gradient: The new line must also have \(m = 4\).
- Form the new equation (\(y = mx + c\)):
\(y = 4x + c\) - Find the y-intercept (\(c\)): Substitute the coordinates \((1, -3)\) into the new equation:
\(-3 = 4(1) + c\)
\(-3 = 4 + c\)
\(c = -7\) - Write the final equation:
\(y = 4x - 7\)
6. Perpendicular Lines (Extended Content Only E4.6)
Perpendicular lines meet or cross at a right angle (90°).
The relationship between the gradients of two perpendicular lines is a bit more complex, involving the negative reciprocal.
Rule for Perpendicular Lines:
If Line A is perpendicular to Line B, the product of their gradients is \(-1\).
\[ m_A \times m_B = -1 \quad \text{or} \quad m_B = -\frac{1}{m_A} \]
How to find the Negative Reciprocal:
- Flip the fraction (Reciprocal).
- Change the sign (Negative).
Example 1:
If \(m_A = 3\), the perpendicular gradient \(m_B\) is:
- (Flip 3/1): 1/3
- (Change sign): \(m_B = -1/3\)
Example 2:
If Line A has the equation \(2y = 3x + 1\), find the gradient of a perpendicular line.
- Rearrange A into \(y = mx + c\):
\[ y = \frac{3}{2}x + \frac{1}{2} \]
\(m_A = 3/2\) - Find the negative reciprocal:
Flip \(3/2\) to get \(2/3\).
Change the sign to get \(-2/3\). - The perpendicular gradient is:
\(m_{\text{perp}} = -2/3\).
Key Takeaway
Parallel gradients are EQUAL. Perpendicular gradients are NEGATIVE RECIPROCALS.