Introduction to Solving Equations and Real-Life Modelling

Welcome! In this chapter, we move beyond just looking at functions as shapes on a graph and start using them to solve problems. Whether it is predicting how a population grows or finding the exact moment a ball hits the ground, we use equations to find the answers. We will explore how to solve these equations by hand (analytically) and using your Graphic Display Calculator (GDC), which is an essential skill for your IB exams.

1. Solving Equations: Analytic vs. Graphic Approaches

In the IB DP Mathematics: Analysis and Approaches course, you need to be comfortable with two ways of finding a solution for \(x\):

Analytic Approach (Paper 1 Style)

This means solving "by hand" using algebra. You use this when the equation is straightforward, such as linear, quadratic, or simple exponential equations.
Example: To solve \(2x + 3 = 7\), you subtract \(3\) and divide by \(2\) to find \(x = 2\).
Detailed methods for specific functions are covered in their respective chapters (like Quadratic Functions or Exponential and Logarithmic Functions).

Graphic Approach (Paper 2 & 3 Style)

Sometimes, an equation is too messy to solve with algebra (for example, \(e^x = x^2 + 3x\)). In these cases, you must use your GDC. There are two main ways to do this on your calculator:

  1. The Intersection Method: Graph \(y_1 = f(x)\) and \(y_2 = g(x)\). Use the "Intersect" function to find the \(x\)-coordinate where they cross.
  2. The Root Method: Rearrange your equation to equal zero (e.g., \(f(x) - g(x) = 0\)). Graph this new function and find the zeros (the \(x\)-intercepts).

Quick Review: Remember that when you solve \(f(x) = g(x)\), the solution is the \(x\)-value. Don't accidentally give the \(y\)-coordinate as your final answer!

2. Intersections and Solutions

When you are asked to find the points of intersection of two curves, you are looking for the coordinates \((x, y)\) that satisfy both equations at the same time.
Key Rule: On Paper 2, if you can't solve it easily with algebra, don't waste time! Sketch the graphs on your GDC and find the intersection points immediately.

Common Mistake to Avoid: When sketching a graph from your GDC onto your paper, ensure you label your axes and include key features like intercepts and asymptotes. The IB examiners want to see that you understand what the calculator is showing you.

3. Real-Life Modelling

Modelling is the process of taking a real-world situation and describing it with a mathematical function.
Analogy: Think of a model like a map. A map isn't the actual city, but it’s a representation that helps you navigate and predict where things are.

The Modelling Process:

  1. Define the Variables: Identify what \(x\) and \(y\) represent (e.g., \(t\) for time in years, \(P\) for population).
  2. Create/Identify the Function: Use the information given to pick the right "shape" (linear, quadratic, exponential, etc.).
  3. Solve the Equation: Find the value of the variable that answers the question (e.g., "When will the population reach 5000?").
  4. Interpret the Result: Does your answer make sense? If you calculate that a ball hits the ground at \(t = -2\) seconds, you know that solution is not valid in a real-world context.

Did you know? Models are rarely "perfect." In your exams, you might be asked to comment on the validity of a model. If a model predicts a human's height will keep growing forever, it is only valid for a certain domain (the age of the person).

4. Advanced Solving: Inequalities and Modulus (HL Only)

For students taking the Higher Level (HL) course, you need to go a step further with inequalities and modulus functions.

Solving \(g(x) \geq f(x)\)

To solve an inequality like this graphically:

  1. Graph both \(y = g(x)\) and \(y = f(x)\).
  2. Find the points where they intersect.
  3. Look for the intervals on the \(x\)-axis where the graph of \(g(x)\) is above (or on) the graph of \(f(x)\).

Modulus Equations \(|f(x)| = g(x)\)

The modulus (or absolute value) \(|x|\) turns everything inside it positive. To solve these:
Analytically: Usually involves solving two cases: \(f(x) = g(x)\) and \(f(x) = -g(x)\).
Graphically: Graph \(y = |f(x)|\) and \(y = g(x)\) and find the intersections.
Note: Remember that \(y = |f(x)|\) reflects any part of the graph below the \(x\)-axis to be above it.

5. Summary Checklist

Key Takeaways:

  • Analytical: Use algebra for Paper 1 (no calculator).
  • Graphical: Use "Intersection" or "Zeros" on your GDC for Paper 2.
  • Modelling: Always check if your answer makes sense in the context of the problem (e.g., no negative time or negative distances).
  • HL Students: Be ready to solve \(g(x) \geq f(x)\) by identifying which graph is higher than the other.

Don't worry if this seems tricky at first! The more you practice using your GDC to "see" the functions, the more intuitive these problems will become. Always start by identifying what the question is asking: are you finding an \(x\)-value (solving) or describing a relationship (modelling)?