1The idea in one paragraph

The graph of a function f is every point (x, y) whose y-coordinate is f(x), so its equation is y = f(x). A point is on the graph exactly when its coordinates satisfy the equation. Putting a graph on paper comes in two strengths. To draw is to be accurate: a scale, plotted points, a ruler for straight lines, a smooth curve. To sketch is to be informative: the right shape in the right place with the features that matter labelled, and no scale needed. On Paper 2 the GDC does the plotting, and your job is to choose a window that shows the whole story, then copy the story onto paper. Graphs can also be added or subtracted: at each x you add or subtract the heights, and a GDC will draw the result for you.

2The graph of a function, and the equation y = f(x)

For each input x in the domain, f gives one output f(x). Plot the point (x, f(x)) for every input and the points join into a curve. That curve is the graph, and its equation is y = f(x): the y-coordinate of every point on it is f of its x-coordinate.

This gives a precise test.

A point (a, b) lies on the graph of y = f(x) exactly when f(a) = b.

Take y = x² − 2.

The same idea runs backwards, and it is behind many of the unknown-constant questions on Paper 1: if you are told a point lies on a graph, substitute it and you have an equation.

3"Draw" and "sketch" are different instructions

The IB's command terms are defined, and the guide expects you to know the difference between these two.

Draw asks for accuracy. Work in pencil and to scale, plot each point where it belongs, rule any straight line, join curved graphs with one smooth curve, and label it.

Sketch asks for the idea. The shape and the relationship must be right, and the relevant features must be shown and labelled, but nothing has to be to scale.

Figure 1 puts the two side by side for y = x² − 4x + 1.

In panel (a) the graph is drawn. There is a scale on both axes, the domain −1 ≤ x ≤ 5 is respected, the seven points from a table of values are plotted, and they are joined by one smooth curve, not by straight segments. In panel (b) the graph is sketched. There is no scale, but the shape is right (a U-shaped parabola), it sits in the right place relative to the axes, and every relevant feature is labelled with its coordinates: the y-intercept (0, 1), the two x-intercepts (0.268, 0) and (3.73, 0), and the vertex (2, −3).

What the marker is looking for differs too. A drawn graph earns marks for accuracy: points in the right squares, a smooth curve, the correct domain. A sketch earns marks for shape and features: correct general shape, correct position, and labelled intercepts, turning points, end points and asymptotes. A beautiful curve with no labels scores badly on a sketch; an unlabelled sketch is not an answer.

Two related words sit nearby. Plot means mark the positions of points on a diagram; you plot before you draw. Label means name the features, usually with coordinates or an equation.

4From screen to paper

On Paper 2 you will usually see the graph on your GDC first. The screen is a window onto the graph: a rectangle from some x-minimum to some x-maximum and some y-minimum to some y-maximum. The guide makes the point that changing this window shows more or less of the function, and a bad window can hide exactly what the question is about.

Figure 2 shows the danger with y = x³ − 12x + 30. In a standard window of −10 ≤ x ≤ 10 and −10 ≤ y ≤ 10, all you see is a steep line crossing the x-axis near −4. The two turning points are at y = 46 and y = 14, far above the top of the screen. Change the window to −6 ≤ x ≤ 6 and −20 ≤ y ≤ 60 and the whole shape appears.

Choosing a window. If the question gives a domain, such as −2 ≤ x ≤ 3, use it for x. Then choose y so that everything shows: many GDCs have a zoom-fit or auto option that sets y for the x-range you chose, and a table of values tells you how high and low the function goes. If there is no domain, zoom out until you are confident nothing is happening off screen, then zoom back in to a window that shows every feature.

Transferring the graph. With a good window, work through the same list every time.

1. Draw and label both axes, x and y (or t and h, or whatever the variables are).
2. Mark the scale lightly with the window limits, so the sketch has the right proportions.
3. Find each key feature with the GDC: y-intercept, x-intercepts, maximum and minimum points, end points of the domain, and any asymptotes. How to find them is 2.4.
4. Mark each feature as a point and write its coordinates beside it, to 3 significant figures unless exact values are obvious.
5. Join the points with the shape you saw on the screen: smooth, with turning points rounded, not pointed.
6. Stop at the end points of the domain, and make an asymptote look like one: the curve gets closer and closer to the line without crossing it (unless the screen shows that it does).

Figure 3 is the result for y = x³ − 12x + 30. The GDC gave the local maximum (−2, 46), the local minimum (2, 14), the y-intercept (0, 30) and the one x-intercept (−4.35, 0). Nothing else needed to be marked.

A sketch is marked on shape and on labelled features. Every intercept, turning point, end point and asymptote on the screen should be on the paper, with coordinates or an equation beside it.

5A sketch from information or a context

Sometimes there is no formula. The question gives you facts, or describes a situation, and asks for a sketch. The method is the same: turn each fact into a feature, place the features, and join them with a sensible shape.

From a description. A cup of coffee is poured at 90 °C into a room at 20 °C. It cools quickly at first, then more and more slowly, getting ever closer to room temperature. Sketch its temperature T against time t.

Read the features out of the words:

• At t = 0, T = 90, so the graph starts at (0, 90) on the vertical axis.
• It decreases for all t, since the coffee only ever cools.
• It falls quickly at first, so it starts steep, then flattens.
• It approaches 20 without reaching it, so the line T = 20 is a horizontal asymptote, drawn dashed and labelled with its equation.
• Time cannot be negative, so the graph starts at t = 0 and there is nothing to its left.

Figure 4 is that sketch. The formula behind it would be something like T = 20 + 70e^(−0.08t), an exponential model you meet in 2.9, but the question did not need it: every mark is for a feature the words described.

From a list of facts. A question might say: f has domain −3 ≤ x ≤ 5, f(−3) = 0, a maximum at (0, 4), a zero at x = 3, a minimum at (4, −2), and f(5) = −1. Plot those five points first. Then join them in order with a smooth curve that rises to the maximum, falls through the zero to the minimum, and rises to the end point. Mark the end points as solid dots, because they are included in the domain. You now have a sketch that satisfies every condition, and that is all the question asked. Q2 below asks you to do exactly this.

Contexts with straight pieces. Not every real graph is curved. Anything that changes at a constant rate, stops, then changes at another constant rate is made of straight segments with corners. Think of a bath filling, standing, and draining. Draw each piece with a ruler, and make sure the gradient of each piece matches its speed: filling slowly is a gentle slope, draining fast is a steep one.

6Sums and differences of functions

Two functions can be added or subtracted to make a new one: (f + g)(x) = f(x) + g(x), and (f − g)(x) = f(x) − g(x). On a graph that means one thing.

At every x, the height of f + g is the height of f plus the height of g.

Figure 5 shows it for f(x) = x² and g(x) = −2x + 3. At x = 2, f(2) = 4 and g(2) = −1, so (f + g)(2) = 4 + (−1) = 3. Do that at every x and you get the amber curve, which is y = x² − 2x + 3. Where g is negative, the sum sits below f; where g is positive, it sits above.

The guide asks you to graph sums and differences using technology, and every GDC does it the same way. Enter f as the first function and g as the second, then define the third as the first plus (or minus) the second, using the calculator's own function names rather than retyping both formulas. That keeps the three graphs linked: change f and the sum updates.

The most useful difference in practice is profit = revenue − cost. Suppose a small workshop that makes and sells x items a day has daily revenue R(x) = 40x − 0.5x² dollars and daily cost C(x) = 200 + 10x dollars. Then its profit is

Figure 6 graphs all three. The profit curve is the vertical gap between R and C. It is positive only where the revenue curve is above the cost line, between the two crossing points. Those are the break-even points, where P(x) = 0. The GDC's zero tool on P gives x = 7.64 and x = 52.4, and its maximum tool gives the highest point of P at (30, 250). So the workshop makes a profit when it sells between 8 and 52 items a day, since x is a whole number of items, and the most it can make is $250 a day, by selling 30.

Notice what the difference graph did. Solving R(x) = C(x) is the same as solving R(x) − C(x) = 0, so the points where two graphs cross are exactly the zeros of their difference. That is one of the two ways 2.4 finds intersections with technology.

7Labels and axes: what every graph needs

The guide says all axes and key features should be labelled. On a sketch that list is short and fixed.

Coordinates from a GDC go to 3 significant figures unless a question says otherwise. If a value is exact and obvious, such as (0, 30), write it exactly.