Non-linear Graphs, Logarithmic Equations and Series (Extended)

Introduction to Thinking with Models (Extended)

Welcome to one of the most powerful chapters in the IB MYP Mathematics framework! In previous chapters, you looked at linear and quadratic models. However, the real world isn't always a straight line or a simple curve. Sometimes things grow incredibly fast (like a viral video) or level off toward a limit. This chapter focuses on Logarithms, Rational Graphs, and Series—the tools mathematicians use to model complex reality.

Quick Review: Before diving in, remember that a function is a relationship where every input has one output. In this chapter, we are looking at "non-linear" functions, which means their graphs are not straight lines.

1. Logarithms: The Power of the Unknown

Logarithms might look intimidating, but they are just a different way of writing exponents. If an exponent asks, "What do I get when I raise 2 to the power of 3?", a logarithm asks, "To what power must I raise 2 to get 8?"

Defining the Logarithm

The relationship between exponents and logarithms is:
\( b^x = y \) is the same as \( \log_b y = x \)

In this expression, \( b \) is the base.
Example: Since \( 10^2 = 100 \), we can say \( \log_{10} 100 = 2 \).

The Laws of Logarithms (Extended)

To solve complex equations, you need to know the "rules of the road." These laws allow you to combine or split logarithms:

• Product Law: \( \log_b(m \times n) = \log_b m + \log_b n \)
• Quotient Law: \( \log_b(\frac{m}{n}) = \log_b m - \log_b n \)
• Power Law: \( \log_b(m^k) = k \times \log_b m \)

Common Mistake to Avoid: A very common error is thinking that \( \log(m + n) \) is the same as \( \log m + \log n \). It is not! The addition happens outside the logs when the terms are multiplied inside.

Logarithmic Functions and Graphs

The graph of \( f(x) = \log_b x \) has a very specific shape.
1. It always passes through the point \( (1, 0) \) because any base to the power of 0 is 1.
2. It has a vertical asymptote at \( x = 0 \). This means the graph gets closer and closer to the y-axis but never actually touches it.
3. As \( x \) gets bigger, the graph grows, but it grows very, very slowly.

Key Takeaway: Logarithms are the "inverse" of exponential functions. If you reflect an exponential graph over the line \( y = x \), you get a logarithmic graph!

2. Rational Functions: Dealing with Divisions

A rational function is a function where one algebraic expression is divided by another, such as \( f(x) = \frac{1}{x} \).

Asymptotes and Behavior

Rational graphs are famous for having "breaks" in them. These are caused by asymptotes.

• Vertical Asymptote: This occurs where the denominator is zero (because we cannot divide by zero!). For \( f(x) = \frac{1}{x} \), the vertical asymptote is \( x = 0 \).
• Horizontal Asymptote: This shows what happens to \( y \) as \( x \) becomes a huge number. For \( f(x) = \frac{1}{x} \), as \( x \) gets massive, \( y \) gets closer to \( 0 \).

Did you know? Rational functions are used in science to model "Inverse Proportion." For example, the more people you have helping to paint a house, the less time (\( y \)) it takes to finish (\( x \)).

3. Arithmetic and Geometric Series (Extended)

In the chapter "Patterns, Sequences and Algorithms," you learned how to find the next number in a list. A Series is what you get when you add those numbers together.

Arithmetic Series

An arithmetic series comes from a sequence where you add a constant difference (\( d \)) each time. To find the sum of the first \( n \) terms (\( S_n \)), use this formula:
\( S_n = \frac{n}{2}(2u_1 + (n - 1)d) \)
Where \( u_1 \) is the first term and \( n \) is the number of terms.

Geometric Series

A geometric series comes from a sequence where you multiply by a constant ratio (\( r \)) each time.
The Finite Sum: \( S_n = \frac{u_1(r^n - 1)}{r - 1} \) (used when you sum a specific number of terms).
The Infinite Sum: If the ratio \( r \) is between -1 and 1 (i.e., \( |r| < 1 \)), the numbers get so small that the sum eventually settles on a single value!
The formula for a sum to infinity is: \( S_{\infty} = \frac{u_1}{1 - r} \).

Example: Imagine you eat half a pizza, then half of what's left, then half of that... theoretically, if you did this forever, you would have eaten exactly 1 whole pizza. That is an infinite geometric series where \( u_1 = 0.5 \) and \( r = 0.5 \).

4. Transformations of Functions (Extended)

In Extended Mathematics, you are expected to apply transformations (shifts and stretches) to exponential, logarithmic, and rational graphs.

If you have a base function \( f(x) \):

• \( f(x) + k \): Moves the graph up by \( k \) units.
• \( f(x - h) \): Moves the graph right by \( h \) units. (Remember: it's the opposite of the sign!).
• \( a \times f(x) \): Stretches the graph vertically by a factor of \( a \).
• \( f(-x) \): Reflects the graph over the y-axis.

Quick Review Box:
- Logarithms solve for the exponent.
- Rational functions have asymptotes where the denominator is zero.
- Arithmetic Series = Adding terms with a constant difference.
- Geometric Series = Adding terms with a constant ratio.
- Infinite Sums only exist if the ratio \( r \) is between -1 and 1.

5. Real-Life Applications (Criterion D)

In your eAssessment, you might be asked to apply these models to real-world scenarios.
- Logarithms are used to measure Earthquakes (Richter Scale) or Sound (Decibels).
- Geometric Series are used to calculate compound interest or the total distance a bouncing ball travels.
- Rational Functions model how the concentration of a medicine in the bloodstream changes over time.

Top Tip for Exams: When working with real-life models, always check if your answer makes sense. If you are calculating the number of people in a room using a logarithmic model and get \( 12.7 \), remember to round to 13 or 12 depending on the context!