Refresh on Graphs and Math Skills

Understanding Graphs

Definition

• A graph is a mathematical representation of a set of objects, called vertices (or nodes), that are connected by links called edges.

Types of Graphs

1. Undirected Graph: In this type, edges have no direction. The connection between the vertices is two-way.

2. Directed Graph (Digraph): Edges have a direction, indicated by arrows, showing the relationship from one vertex to another.

3. Weighted Graph: Edges have weights, often representing cost, distance, or time.

4. Unweighted Graph: Edges are simply connections without any associated weights.

5. Cyclic Graph: Contains at least one cycle (a path from a node back to itself).

6. Acyclic Graph: Does not contain cycles.

   • Tree: A special type of acyclic graph with a hierarchical structure.

Graph Representation

• Adjacency Matrix: A square matrix represents a graph, where each cell indicates whether pairs of vertices are adjacent.

• Adjacency List: A collection of lists or arrays, one for each vertex, with each list containing the adjacent vertices.

Properties of Graphs

• Degree: The number of edges connected to a vertex (in-degree and out-degree in directed graphs).

• Path: A sequence of edges that connect a sequence of vertices.

• Connected Graph: There is a path between every pair of vertices.

• Disconnected Graph: At least two vertices do not have paths between them.

Applications

• Computer networks, social network analysis, operations research, and various algorithmic processes rely heavily on graph theory.

Common Algorithms

1. Depth-First Search (DFS): A traversal algorithm for exploring nodes and edges of a graph.

2. Breadth-First Search (BFS): Another traversal algorithm exploring all neighbors at the present depth before moving on to nodes at the next depth level.

3. Dijkstra's Algorithm: A shortest path algorithm for weighted graphs.

4. An Algorithm*: Used for finding the shortest path in a graph with heuristics.

5. Kruskal's and Prim's Algorithms: For finding the minimum spanning tree of a weighted graph.

Examples of Different Graphs

Examples of Different Graphs

1. Undirected Graph

• Vertices (Nodes): A, B, C, D

• Edges: {(A, B), (A, C), (B, C), (C, D)}

2. Directed Graph

• Vertices (Nodes): 1, 2, 3, 4

• Edges: {1 -> 2, 2 -> 3, 3 -> 4, 1 -> 4}

3. Weighted Graph

• Vertices (Nodes): P, Q, R

• Edges: {(P, Q, weight=5), (Q, R, weight=3), (P, R, weight=10)}

4. Cyclic Graph

• Vertices (Nodes): X, Y, Z

• Edges: {(X, Y), (Y, Z), (Z, X)}

5. Tree

• Vertices (Nodes): A, B, C, D, E

• Edges: {(A, B), (A, C), (B, D), (B, E)}

6. Acyclic Graph

• Vertices (Nodes): 1, 2, 3, 4

• Edges: {(1, 2), (1, 3), (2, 4)}

Axes Labels

• X-axis: Typically represents one variable (e.g., time, distance).

• Y-axis: Represents another variable (e.g., cost, value).

In graphing, intercepts refer to the points where a graph intersects the axes. There are two key types of intercepts:

1. X-Intercept: This is the point where the graph crosses the x-axis. At this point, the value of y is zero.

2. Y-Intercept: This is the point where the graph crosses the y-axis. At this point, the value of x is zero.

These intercepts can be helpful in understanding the behavior of a function and sketching its graph.

Components of Graphing

1. Variables: Represent the quantities being measured or compared in a graph. They are typically represented on the axes (e.g., x and y).

2. Labels: Provide information about the data represented on the axes or in the graph. They help clarify what each axis represents and may include units of measurement.

3. Numbers and Tick Marks: Indicate scale and intervals on the axes. Tick marks are small lines or markers on the axes that represent specific values.

4. Origin: The point where the axes intersect (0,0). It serves as the reference point for all other points in the graph.

5. Data Points: The individual points plotted on the graph that represent the values of the variables being analyzed.

6. A Line: In graphing, it typically represents a linear relationship between the variables. It can connect data points or represent a function.

7. A Curve: Represents a non-linear relationship, showing how changes in one variable affect another. Curves can vary in shape depending on the relationship between the variables.

Slopes and Lines

Slope: The slope of a line is a measure of the steepness or incline of the line. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

Slope of a Line: The slope can be represented mathematically as:

[ \text{slope} (m) = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1} ]

Where (x1, y1) and (x2, y2) are two distinct points on the line. A positive slope means the line is increasing, a negative slope indicates it's decreasing, while a slope of zero represents a horizontal line.

Rise and Run:

• Rise: The vertical change between two points on the line.

• Run: The horizontal change between the same two points.

The slope of a line can be classified as:

• Positive Slope: Indicates that as the x-values increase, the y-values also increase, resulting in an upward incline when graphed.

• Negative Slope: Indicates that as the x-values increase, the y

Types of Slopes

• Negative Slope: An example of a negative slope can be seen in a function where, as the x-values increase, the y-values decrease. For instance, the line represented by the equation y = -x will have a negative slope, slanting downward from left to right.

• Positive Slope: A positive slope occurs when, as the x-values increase, the y-values also increase. An example is the line represented by the equation y = x, which slopes upward from left to right.

• Slope of Zero: A line with a slope of zero is horizontal, meaning it does not rise or fall as it extends across the x-axis. An example is the equation y = 3, which indicates that y always equals 3, regardless of x.

• Same Slope: Two lines that are parallel will have the same slope. For instance, the lines y = 2x + 1 and y = 2x - 3 both have a positive slope of 2, indicating they are parallel and will never intersect.

An infinite slope is typically represented by a vertical line on a graph. In terms of slope calculation, the slope formula (rise/run) would involve a division by zero (since the run would be zero), which mathematically corresponds to an undefined or infinite slope. For example, the line described by the equation x = a (where 'a' is a constant) is vertical and has an infinite slope.

• Demand Curve: A line on a graph that shows how much of something people want to buy at different prices. It usually goes down because people buy more when prices are lower.

• Supply Curve: A line on a graph that shows how much of something producers are willing to sell at different prices. It usually goes up because producers make more when prices are higher.

• Green Line: This can refer to different ideas, but it often shows where supply and demand balance each other or relates to environmental topics.

• Quantity Demanded: The amount of a product that people want to buy at a certain price.

• Quantity Supplied: The amount of a product that producers are ready to sell at that same price.

• Surplus: When there is too much of a product for sale than people want to buy, leading to leftover items.

• Shortage: When people want to buy more of a product than what is available for sale, so there isn’t enough for everyone who wants it.

Tangent Lines and the Slope Along a Curve

Tangent Line

• A tangent line is a straight line that touches a curve at a single point without crossing it.

• The slope of the tangent line represents the instantaneous rate of change of the curve at that point.

Slope Along a Curve

• The slope of a curve can vary from point to point.

• At any given point on the curve, the slope can be found using calculus, particularly through the use of derivatives.

• The derivative of a function at a certain point gives the slope of the tangent line to the curve at that point.

Mathematical Representation

• The slope of the curve at point (x, f(x)) can be defined as:

[ m = f'(x) ]

Where ( f'(x) ) is the derivative of the function, giving the slope of the tangent line at that point.

Importance

• Tangent lines and their slopes are vital in various applications including physics for understanding motion, and in economics for evaluating cost and revenue functions.

Area Formulas

1. Square: The area (A) of a square is calculated using the formula:

   [ A = s^2 ]

   Where ( s ) is the length of one side of the square.

2. Rectangle: The area (A) of a rectangle is calculated using the formula:

   [ A = l \times w ]

   Where ( l ) is the length and ( w ) is the width of the rectangle.

3. Triangle: The area (A) of a triangle is calculated using the formula:

   [ A = \frac{1}{2} \times b \times h ]

   Where ( b ) is the length of the base and ( h ) is the height of the triangle.

4. Area of a quadrilateral: General Quadrilateral: A = 0.5 d1*d2 * sin(θ) (where d1 and d2 are the diagonals and θ is the angle between them)

5. Rectangle: A = l * w

6. Square: A = s * s

7. Parallelogram: A = b * h (where b is the base and h is the height).

To find the percentage change without using fractions, you can follow these steps:

1. Identify the Original Value and New Value: Let’s say the original value is Original and the new value is New.

2. Calculate the Change:

   Change = New - Original

3. Calculate the Percentage Change:

   Percentage Change = (Change ÷ Original) × 100%

   You can express this formula in words:

   • Subtract the original value from the new value to find the change.

   • Divide the change by the original value to find the ratio of change.

   • Multiply this result by 100 to convert it into a percentage.

4. Interpret the Result: If the percentage change is positive, it means there was an increase; if it's negative, there was a decrease.

Absolute Change

Absolute change refers to the actual difference between two values. It is calculated by subtracting the original value from the new value.

Formula:

Absolute Change = New Value - Original Value

Relative Change

Relative change is a measure of how much a value has changed in relation to the original value. It expresses this change as a percentage of the original value.

Formula:

Relative Change = (Change ÷ Original) × 100% Relative Change can also be expressed as: Relative Change = ((New Value - Original Value) ÷ Original Value) × 100%

Note:

• Absolute change provides the straightforward numerical difference between two values, while relative change shows this difference as a percentage of the original value, providing context for the magnitude of the change.