Mathematics (Specification A) | Graphs

Welcome to the Chapter on Graphs!

Hello there! This chapter, "Graphs," is one of the most visual and useful parts of the Mathematics curriculum. It bridges the gap between abstract algebra (equations) and concrete pictures (visual plots).

Why is this important? Graphs help us understand relationships quickly – whether it's speed over time, how much money you earn based on hours worked, or predicting the trajectory of a ball. Don't worry if drawing straight lines seems basic; we will quickly move on to complex curves and powerful problem-solving techniques!

Section 1: The Coordinate Plane – Your Map for Maths

1.1 Plotting Coordinates

Before we draw lines, we need to know how to plot points. The area where we draw our graphs is called the Cartesian Plane.

• It consists of two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical).
• The point where they cross is the Origin, \( (0, 0) \).
• A point is written as an ordered pair \( (x, y) \).

Memory Aid: Always go Across first (X), then Up or Down (Y). Think of it like walking along the corridor (X) before taking the lift (Y).

Example: To plot \( (3, -2) \), start at the Origin, move 3 units right (positive X), and then 2 units down (negative Y).

Section 2: Straight Line Graphs

2.1 Drawing Straight Lines Using a Table of Values

The simplest way to draw any graph, especially if you are unsure, is to create a table of values.

Step-by-Step Method:

1. Choose X values: Pick a range of x-values (usually from -3 to 3 is a good start).
2. Calculate Y values: Substitute each chosen x-value into the equation to find the corresponding y-value.
3. Plot Points: Plot the resulting coordinate pairs \( (x, y) \).
4. Connect: Use a ruler to draw a single straight line through all the plotted points, extending it across the grid.

Common Mistake to Avoid: Never draw little segments connecting the dots. A straight line graph should be one long, continuous line with arrows on the ends (if appropriate) to show it continues forever.

2.2 The Equation of a Straight Line: \( y = mx + c \)

This formula is essential. If you can understand the two main parts, you can draw any straight line without needing a huge table of values!

The equation \( y = mx + c \) tells us two crucial things:

1. \( m \) is the Gradient (Steepness)
2. \( c \) is the Y-intercept (Where it crosses the Y-axis)

The Y-Intercept (\( c \))

The y-intercept (\( c \)) is the point where the line crosses the vertical y-axis. It is always the coordinate \( (0, c) \).

Example: In the equation \( y = 3x - 5 \), the y-intercept is -5. The line crosses the y-axis at \( (0, -5) \).

The Gradient (\( m \))

The gradient (\( m \)) measures the steepness and direction of the line.

• Positive gradient (\( m > 0 \)): Line slopes up from left to right.
• Negative gradient (\( m < 0 \)): Line slopes down from left to right.

2.3 Calculating the Gradient (\( m \))

The gradient is calculated using the formula: $$ m = \frac{\text{Change in } y}{\text{Change in } x} = \frac{\text{Rise}}{\text{Run}} $$

If you are given two points, \( (x_1, y_1) \) and \( (x_2, y_2) \), the formula becomes: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$

Step-by-Step Example: Find the gradient between A(1, 4) and B(5, 12).

1. Label your points: \( x_1 = 1, y_1 = 4 \) and \( x_2 = 5, y_2 = 12 \).
2. Substitute into the formula: $$ m = \frac{12 - 4}{5 - 1} $$
3. Calculate: $$ m = \frac{8}{4} = 2 $$ The gradient is 2.

2.4 Special Straight Lines

There are two special cases you must know:

• Horizontal Lines: These lines have zero gradient (\( m = 0 \)). Their equation is always in the form \( y = c \). Example: \( y = 4 \) is a horizontal line crossing the y-axis at 4.
• Vertical Lines: These lines have an undefined (infinite) gradient. Their equation is always in the form \( x = a \). Example: \( x = -1 \) is a vertical line crossing the x-axis at -1.

2.5 Parallel and Perpendicular Lines

Parallel Lines

Parallel lines never cross and are equally steep. Therefore, they must have the same gradient (\( m \)).

Example: \( y = 5x + 1 \) and \( y = 5x - 8 \) are parallel because both gradients are \( m=5 \).

Perpendicular Lines

Perpendicular lines meet at a right angle (\( 90^\circ \)). If the gradient of the first line is \( m_1 \), the gradient of the perpendicular line, \( m_2 \), is the negative reciprocal.

$$ m_2 = - \frac{1}{m_1} $$

Trick: To find the negative reciprocal, you must flip the fraction and change the sign.

Example 1: If \( m_1 = 4 \) (or \(\frac{4}{1}\)), then \( m_2 = -\frac{1}{4} \).
Example 2: If \( m_1 = -\frac{2}{3} \), then \( m_2 = +\frac{3}{2} \).

Did you know? Two lines are perpendicular if and only if the product of their gradients is -1: \( m_1 \times m_2 = -1 \).

Section 3: Non-Linear Graphs (The Curves)

Non-linear graphs produce curves, not straight lines. You will draw these the same way you draw straight lines: by using a comprehensive table of values, but plotting MUST be done carefully and joined with a smooth curve (NO rulers!).

3.1 Quadratic Graphs (\( y = ax^2 + bx + c \))

Quadratic equations contain an \( x^2 \) term (but no higher powers). Their graphs are called Parabolas.

• If the coefficient of \( x^2 \) is positive (e.g., \( y = x^2 + \dots \)), the graph is U-shaped (a "Smile"). It has a minimum point.
• If the coefficient of \( x^2 \) is negative (e.g., \( y = -x^2 + \dots \)), the graph is inverted U-shaped (a "Frown"). It has a maximum point.

Critical Feature: The turning point (maximum or minimum). Ensure your table of values includes the coordinates around this point so you can draw the curve accurately.

3.2 Cubic Graphs (\( y = ax^3 + \dots \))

Cubic equations contain an \( x^3 \) term. Their graphs generally have an S-shape or a snake-like curve.

• They typically pass through the x-axis up to three times.
• They often have two turning points (a local maximum and a local minimum), though sometimes they just have a point of inflection (a flattened section).

Tip for Plotting: Since cubic numbers grow very fast, choose your x-values carefully (e.g., from -2 to 2) and make sure your y-axis scale can handle the large numbers you calculate.

3.3 Reciprocal Graphs (\( y = \frac{a}{x} \))

The simplest reciprocal function is \( y = \frac{k}{x} \) (where \( k \) is a constant). These graphs are called Hyperbolas.

Crucial Point: You can never divide by zero!

• When \( x = 0 \), y is undefined. This means the graph never touches the y-axis.
• As \( x \) gets very large (positive or negative), \( \frac{k}{x} \) gets very close to zero, meaning the graph never touches the x-axis.

These "never-touch" lines are called Asymptotes. For \( y = \frac{k}{x} \), the asymptotes are the x-axis and the y-axis. The graph consists of two separate, mirror-image curves in opposite quadrants (Q1 and Q3, or Q2 and Q4).

Section 4: Using Graphs to Solve Equations

One of the most powerful uses of graphs is solving equations, especially when algebra is difficult or impossible.

4.1 Solving Simultaneous Equations Graphically

When you are asked to solve two simultaneous equations, you are looking for the point \( (x, y) \) that satisfies both equations at the same time.

Graphical Method:

1. Draw the graph of the first equation accurately.
2. Draw the graph of the second equation accurately on the same set of axes.
3. The solution is the point(s) of intersection (where the lines/curves cross).

Example: If the lines \( y = 2x + 1 \) and \( y = -x + 4 \) cross at the point \( (1, 3) \), then the solution to the simultaneous equations is \( x=1 \) and \( y=3 \).

4.2 Solving Equations by Finding Intersections

We can use graphs to find the approximate solutions (roots) of a single, complex equation, especially non-linear ones.

If you have an equation like \( x^2 - 3x - 1 = 0 \), you can find its solutions by looking at where the graph of \( y = x^2 - 3x - 1 \) crosses the x-axis (because at these points, \( y=0 \)).

Solving by Finding the Intersection of Two Different Functions

Sometimes, the exam question asks you to solve an equation by drawing a new, simple line.

Step-by-Step Example: Use the graph of \( y = x^2 - 4x + 2 \) (already drawn) to solve the equation \( x^2 - 4x - 1 = 0 \).

1. Compare the Equations:
   Drawn Graph: \( y = x^2 - 4x + 2 \)
   Target Equation: \( x^2 - 4x - 1 = 0 \)
2. Find the Difference: Rearrange the Target Equation to look like the Drawn Graph: $$ x^2 - 4x + 2 - 3 = 0 $$ (We changed -1 to +2, so we must subtract 3).
3. Set up the Intersection: $$ (x^2 - 4x + 2) = 3 $$ Since the drawn graph is \( y = x^2 - 4x + 2 \), we are looking for the solutions where \( y = 3 \).
4. Draw the New Line: Draw the horizontal line \( y = 3 \).
5. Read the Solutions: The x-coordinates where the parabola and the line \( y=3 \) cross are the solutions to \( x^2 - 4x - 1 = 0 \).

This technique is vital for finding approximate solutions when exact algebraic methods are too difficult or not required. Always read the values directly from your graph as accurately as possible.