Introduction to Graphs and Their Key Features

Welcome! In the world of functions, graphs are our most powerful visual tools. Think of a graph as the "portrait" of a function—it shows us how the function behaves, where it changes direction, and how it relates to other functions. Whether you are using a Graphic Display Calculator (GDC) or sketching by hand, understanding these visual signatures is essential for success in IB Mathematics: Analysis and Approaches.

In this chapter, we will focus on how to identify the "landmarks" of a graph and how to use technology to solve complex problems quickly. Don't worry if you find graphs a bit messy at first; once you know what features to look for, they start to tell a very clear story!


Section 1: The "Landmarks" of a Graph

When you look at a graph \(y = f(x)\), there are specific features you must always be able to identify. These are the "key features" referred to in the syllabus.

1. Axis Intercepts

  • The \(y\)-intercept: This is where the graph crosses the vertical axis. It occurs when \(x = 0\). To find it algebraically, calculate \(f(0)\).
  • The \(x\)-intercepts (Roots or Zeros): These are the points where the graph crosses the horizontal axis. They occur when \(y = 0\). To find them, solve the equation \(f(x) = 0\).

2. Turning Points (Extrema)

These are points where the function changes from increasing to decreasing, or vice versa.

  • Local Maximum: The "top of a hill" on the graph.
  • Local Minimum: The "bottom of a valley" on the graph.

3. Asymptotes

An asymptote is a line that the graph approaches but never actually reaches. You will often see these as dashed lines on a sketch. Note: You will study these in more detail in the "Reciprocal and Rational Functions" chapter.

4. Points of Intersection

When you have two functions, say \(f(x)\) and \(g(x)\), the points where their graphs cross are the points of intersection. At these specific \(x\)-values, \(f(x) = g(x)\).

Quick Review: Every time you look at a graph, ask yourself: "Where does it hit the axes, where does it turn, and does it approach any invisible lines?"


Section 2: Sketching vs. Drawing

In IB exams, the command terms "Sketch" and "Draw" mean different things. It is a common mistake to confuse them!

  • Draw: This requires a high level of accuracy. You should use a ruler and plot specific points on graph paper with a consistent scale.
  • Sketch: This is a representative drawing. You don't need a perfect scale, but you must label the key features (intercepts, turning points, and asymptotes) with their coordinates or equations. The shape must also be mathematically correct.

How to transfer a graph from GDC to paper:

  1. Use your GDC to view the function.
  2. Identify the window settings (the min and max \(x\) and \(y\) values).
  3. Locate the intercepts and turning points using the Analyze Graph or G-Solv tools.
  4. Draw your axes on paper and mark these key points first.
  5. Connect the points with a smooth curve that matches the shape on your screen.

Did you know? If a question asks you to sketch a graph "in context" (like height vs. time), you usually only need to show the part of the graph where the variables make sense (e.g., time \(t \geq 0\)).


Section 3: Using Technology (GDC)

For Paper 2 and Paper 3, your GDC is your best friend. The IB syllabus expects you to be proficient at using technology to find features that are difficult to calculate by hand.

Solving Equations Graphically

To solve an equation like \(f(x) = k\):

  1. Graph the function \(y_1 = f(x)\).
  2. Graph the horizontal line \(y_2 = k\).
  3. Find the intersection point between the two lines. The \(x\)-coordinate of this point is your solution.

Finding Intersections of Two Curves

To solve \(f(x) = g(x)\):

  1. Graph both \(y_1 = f(x)\) and \(y_2 = g(x)\).
  2. Use the Intersection tool on your GDC.
  3. If there are multiple points where they cross, make sure to find all of them!

Common Mistake: When writing down coordinates from your GDC, make sure you don't round too early. Always use at least 3 significant figures for your final IB answer.


Section 4: Sums and Differences of Functions

The syllabus mentions graphing the sum or difference of two functions: \(h(x) = f(x) \pm g(x)\).

Imagine you have two graphs. At any given \(x\)-value, the new graph \(h(x)\) is simply the result of adding or subtracting the \(y\)-values of the original two graphs.
Example: If \(f(2) = 5\) and \(g(2) = 3\), then the graph of \(f+g\) will have a point at \((2, 8)\).

On your GDC, you can easily do this by entering:
\(y_1 = \text{first function}\)
\(y_2 = \text{second function}\)
\(y_3 = y_1 + y_2\)


Summary Checklist

  • Identify Intercepts: Set \(x=0\) for \(y\)-intercept; set \(y=0\) for \(x\)-intercepts.
  • Find Turning Points: Use GDC to locate local maxima and minima.
  • Intersection Points: Use GDC to find where \(f(x) = g(x)\).
  • Labeling: Always label axes and key points when sketching.
  • GDC Skills: Ensure you know how to adjust the "Window" so you can actually see all the key features of the graph.

Don't worry if this seems tricky at first! The more you practice "exploring" functions on your calculator, the more natural these features will become.


Key Takeaway

A graph is more than just a line; it is a collection of key features. Intercepts, turning points, and intersections are the "answers" to most functional problems in IB Mathematics.