IB Math Applications: A Crash Course for Tomorrow's Test

Let's explore the different types of numbers in mathematics, starting with the broadest set: real numbers.Within real numbers, we have rational numbers, which can be expressed as fractions.Integers form a subset of rational numbers, including all whole numbers and their negatives.And at the core, we have natural numbers, the counting numbers starting from one.Let's look at some examples from each number set.Numbers can be visualized on a number line, showing their relative positions and order.Understanding the properties of mathematical operations is crucial for problem solving.A key skill is converting between fractions, decimals, and percentages.These concepts have many practical applications in everyday life.Linear functions form straight lines and follow the pattern y equals m x plus b.Here's an example where m equals 2 and b equals 1. The slope m determines steepness, while b shifts the line up or down.When we increase the slope, the line becomes steeper.Quadratic functions create parabolas and follow the pattern y equals a x squared plus b x plus c.This parabola has a equals 1 and c equals negative 2, creating an upward-opening parabola shifted down 2 units.We can also write quadratics in vertex form, where h and k represent the vertex coordinates.When we decrease a, the parabola becomes wider.Exponential functions show rapid growth or decay and follow the pattern y equals a times b to the x power.Here's an exponential growth function where b equals 2, showing doubling at each step.When b is less than 1, we get exponential decay, as shown here with b equals one half.Let's explore how we can transform functions. Starting with a basic quadratic, y equals x squared.Adding a constant shifts the function vertically. Adding 2 moves everything up 2 units.Replacing x with x minus h shifts the function h units right.Multiplying by a constant greater than 1 stretches the function vertically.In population growth, exponential functions model how populations increase over time. Here, we start with 1000 organisms and grow by 10 percent each time period.Quadratic functions model projectile motion, showing how objects move through the air. The negative coefficient represents gravity's effect.Linear functions often represent cost models, where the slope shows the rate of change and the y-intercept represents fixed costs.In statistics, we start by understanding measures of central tendency.The mean represents the average of all values, while the median is the middle value when data is ordered.When measuring spread, we look at the interquartile range and standard deviation.In probability, we often work with independent events, where the occurrence of one event doesn't affect the other.Conditional probability examines the likelihood of an event occurring given that another event has occurred.Data can be represented in various ways, with histograms being particularly useful for showing frequency distributions.Let's explore compound interest, which is the foundation of financial growth.The formula shows how money grows when interest is reinvested. Let's compare compound and simple interest for a ten thousand dollar investment at seven percent annual interest.Now, let's examine depreciation, which shows how assets lose value over time.Let's look at a practical example of car depreciation over three years.Finally, let's understand loan calculations, particularly the monthly payment formula.Here's how monthly payments break down for a typical mortgage.In a typical mortgage, a large portion of early payments goes to interest rather than principal.