Introduction to Graphs and Their Features
Welcome! In this chapter, we are going to explore how to visualize functions. If a function is like a set of instructions, then a graph is the "story" those instructions tell. By looking at a graph, we can instantly see where a business makes a profit, when a ball reaches its highest point, or where two different paths cross.
In the IB Applications and Interpretation (AI) course, your Graphic Display Calculator (GDC) is your best friend. We will focus on how to use your technology to find important information and how to transfer what you see on your screen onto your exam paper correctly.
1. Sketching vs. Drawing Graphs
In your exams, the IB uses specific command terms that tell you exactly how much detail you need to provide. Don't worry if this seems picky—it's actually a great way to save time!
The "Sketch"
A sketch doesn't need to be perfectly to scale. It is a representative drawing that shows the general shape of the function and its most important features.
Quick Tip: Always use a pencil for sketches! If you make a mistake, it’s much easier to fix.
The "Draw"
When the exam asks you to draw, it means you need to be precise. You should use a ruler, use the specific scale provided (if there is one), and plot points accurately on graph paper.
Note: For more on specific linear or quadratic shapes, check out the "Modelling with standard functions" chapter.
2. Key Features of Graphs
When you look at any graph, there are "landmarks" you should always look for. These features help us interpret what is happening in a real-world context.
- The \(y\)-intercept: This is where the graph crosses the vertical axis. It happens when \(x = 0\). In real life, this often represents the "starting value" (like the initial cost of a taxi before you even move).
- The \(x\)-intercepts (Roots/Zeros): These are the points where the graph crosses the horizontal axis. This happens when \(y = 0\). For example, if your graph shows the height of a diver, the \(x\)-intercept is when they hit the water.
- Turning Points (Local Maximum and Minimum): These are the "peaks" and "valleys" of your graph.
- A Maximum is the highest point in a certain area.
- A Minimum is the lowest point in a certain area.
- Asymptotes: These are "invisible lines" that a graph gets closer and closer to but never actually touches. You will often see horizontal asymptotes in exponential models or vertical asymptotes in inverse variation models.
Quick Review: To find these features on your GDC, look for the G-Solv (Casio) or Analyze Graph (TI-Nspire) menus!
3. Transferring from GDC to Paper
One of the most common places students lose marks is when they copy a graph from their calculator screen to their exam paper. Even for a simple sketch, you must follow these rules:
The Five Golden Rules of Graph Sketching:
- Label your axes: Always write \(x\) and \(y\), or the specific variables given in the question (like \(t\) for time or \(h\) for height).
- Show the intercepts: Clearly mark where the graph crosses the axes. If the question asks for coordinates, write them in the form \((x, y)\).
- Label key points: If you found a maximum or minimum, mark it!
- Indicate the scale: You don't need a grid, but you should show a few numbers on the axes so the reader knows the "size" of your graph.
- Correct shape: Make sure your curves are smooth and your straight lines are drawn with a ruler.
4. Intersections: Where Paths Meet
An intersection is a point where two different graphs cross each other. Mathematically, this is the point where the two functions are equal.
If you have two functions, \(f(x)\) and \(g(x)\), the intersection is the point where \(f(x) = g(x)\).
How to find intersections using your GDC:
- Enter both functions into your calculator's graphing menu (\(y1\) and \(y2\)).
- Draw/Graph them.
- Use the Intersect tool (usually found in the Analysis or Trace menus).
- The calculator will give you an \((x, y)\) coordinate.
Real-World Example: If \(f(x)\) is the cost of producing items and \(g(x)\) is the money made from selling them, the intersection point is the "Break-Even Point"—the moment you stop losing money and start making a profit!
5. Sums and Differences of Functions
Sometimes, you might be asked to graph a new function that is made by adding or subtracting two other functions, such as \(h(x) = f(x) + g(x)\).
Don't panic! You don't need to do this by hand. Simply type the full expression into your GDC. For example, if \(f(x) = 2x\) and \(g(x) = x^2\), you can just type \(y1 = 2x + x^2\) and let the technology do the heavy lifting.
Did you know? This is often used in physics when "superimposing" waves—adding two sound waves together creates a new, combined sound graph!
Summary Table: Key Terms to Remember
| Term | What it means | How to find it (\(x\) or \(y\)) |
|---|---|---|
| \(y\)-intercept | Where it starts on the vertical axis | Set \(x = 0\) |
| \(x\)-intercept | Where it hits the floor (roots) | Set \(y = 0\) |
| Vertex | The "turn" (Max or Min) | GDC Max/Min tool |
| Intersection | Where two lines cross | GDC Intersect tool |
Key Takeaway: Your GDC is a powerful tool, but it can only show you what's on the screen. Always adjust your Window Settings (the \(x\) and \(y\) limits) to make sure you can see all the key features like intercepts and turning points before you start sketching!