IB Mathematics Analysis and Approaches SL 2.3—graphing Notes

A Graph Turns An Equation Into A Picture You Can Read

1. An equation like $y = x^2 - 3$ is a rule, and a rule on its own is hard to feel: you cannot see at a glance where it is high, where it is low, or where it crosses zero.
2. Draw its graph and all of that becomes obvious in one look, because the shape of the curve carries the same information the algebra does.
3. This is why almost every exam question that involves a function eventually asks you to sketch it, read a value off it, or use it to explain what is happening.
4. On Paper 2 you have a graphing calculator, so producing the picture is quick, and the marks move to a different skill: copying that picture onto paper correctly and labelling it.

The Graph Of A Function Is The Set Of Points (x, f(x))

1. A function takes an input $x$ and returns exactly one output, which we call $f(x)$.
2. To graph it, we pair each input with its output and treat that pair as a coordinate point $(x,\, f(x))$.
3. The equation of the graph is written $y = f(x)$, which simply says the height $y$ of the curve above any $x$ equals the output $f(x)$.
4. Plotting every such point for all allowed $x$ traces out the whole curve: $$ \text{graph of } f = \{\,(x,\, y) : y = f(x)\,\} $$
5. So a point lies on the graph exactly when its coordinates satisfy the equation, and it lies off the graph when they do not.

Every Graph Must Show Its Key Features, Labelled

1. A useful graph is not just a curve: it is a curve with its important points named.
2. The IB guidance for this subtopic is blunt: all axes and all key features must be labelled, on every graph you produce.
3. The key features you should be ready to mark are these.
   1. The axes, each labelled with the variable it represents and a scale.
   2. The y-intercept, where the curve crosses the vertical axis.
   3. The x-intercepts (also called zeros or roots), where the curve crosses the horizontal axis.
   4. Any maximum or minimum turning points.
   5. Any asymptotes, drawn as dashed guide lines the curve approaches.
   6. The endpoints, if the function is only defined on part of the axis.

"Draw" And "Sketch" Are Different Commands

1. In the exam the command word at the start of the instruction tells you how precise your graph must be.
2. Sketch means show the general shape and the key features, labelled but not to scale, done freehand.
3. Draw means produce an accurate, to-scale graph, with points plotted correctly and a ruler or a smooth curve used, as if you were plotting real data.

Creating A Sketch From Given Information Or A Context

1. Often you are not handed a formula to plot but a set of facts, and your job is to build a sketch that fits them all.
2. Work through a fixed order so nothing is missed.
   1. Read off every fact given: intercepts, turning points, asymptotes, endpoints, or a described behaviour.
   2. Set up and label the axes to match the context, choosing a scale that fits the given values.
   3. Mark each given point on the axes before drawing anything.
   4. Join the marked points with a smooth curve of the right shape, respecting the described behaviour between them.
3. In a real-world context, only draw the part of the curve that makes sense, for example time $t \geq 0$.

Transferring A Graph From The Calculator Screen To Paper

1. On Paper 2 you will often graph a function on your calculator and then have to reproduce it on paper as a sketch.
2. The screen shows the shape, but your paper version has to carry the labels, so copy it deliberately.
   1. Match your paper axes to the calculator's viewing window, so the same region of the curve is shown.
   2. Copy the overall shape and the direction the ends of the curve head.
   3. Use the calculator's built-in tools to read the key feature values, then write those coordinates onto your paper sketch.
   4. Label the axes with their variables and a scale that matches the window.
3. Choosing a sensible viewing window matters: too small a window can hide an intercept or a turning point, so widen it until the whole shape is visible before you copy.

Using Technology To Graph Sums And Differences Of Functions

1. Your calculator can graph more than one function at once, and it can graph a combination such as $y = f(x) + g(x)$ directly.
2. The idea behind a sum of functions is that heights add at each input.
   1. At any $x$, the height of $f(x) + g(x)$ is the height of $f$ there plus the height of $g$ there.
   2. So you can build the sum point by point: pick an $x$, add the two $y$-values, and plot the result.
3. A difference $f(x) - g(x)$ works the same way, subtracting the heights instead of adding them.
4. Adding functions is not the same as combining them into a composite functions, where one function is fed into another; a sum simply adds the two outputs.