1The idea in one paragraph

A graph is described by a short list of key features: where it crosses the axes, where it turns, whether it is symmetric, and which lines it approaches far out. Those features are what a sketch must show and what most questions ask for. On Paper 2 your GDC finds each one with a built-in tool (zero, maximum, minimum, intersect), and your job is to know which tool answers which question and to report the result properly. The most powerful of them is the intersection: two graphs meet exactly where their equations are equal, so any equation f(x) = g(x), however hard to solve by algebra, can be solved by looking for where y = f(x) and y = g(x) cross.

2The features, and their names

Figure 1 shows one graph with its features labelled: the cubic y = x³ − 3x² − 9x + 5.

Three names for one idea deserve a sentence. The zeros of a function f, the roots of the equation f(x) = 0, and the x-intercepts of y = f(x) are the same numbers seen three ways: as inputs, as solutions, and as points on a graph. For Figure 1 they are x = −2.18, 0.489 and 4.69. A question asking for the zeros wants the x-values; a question asking for the x-intercepts wants points, (−2.18, 0) and so on.

One more distinction costs marks easily. The maximum value of a function is a y-value, such as 10. The maximum point is a pair of coordinates, (−1, 10). Answer the one you are asked for.

3Using the GDC: what each tool gives you

On Paper 2 you are expected to find these features with graphing technology, and a short note of the method you used helps the marker award method marks. The tools have different names on different calculators but the same jobs:

On a TI-84 these live under CALC; on a Casio under G-Solv; on a TI-Nspire under Analyze Graph. Whichever you use, three habits matter.

Find each feature separately. A cubic with three zeros needs the zero tool three times, with bounds around each one. The tool gives you one answer per use.

Round only at the end, and to 3 significant figures. The GDC shows 0.488872256…; you write 0.489. If that value feeds into a later calculation, use the stored value, not the rounded one, or the next answer drifts.

Write down what you did. A line such as "using GDC: local minimum at (3, −22)" is enough. An answer with no working and no mention of technology can still score, but only if it is exactly right.

On Paper 1, with no calculator, you find the same features by algebra: factorise to get zeros, substitute x = 0 for the y-intercept, read the vertex from a completed square (2.6), and read asymptotes from the form of the function. The features are the same; only the route changes.

4Maximum and minimum values

A local maximum is a point higher than every nearby point on the graph: the top of a hill. A local minimum is the bottom of a valley. For y = x³ − 3x² − 9x + 5 the GDC gives a local maximum at (−1, 10) and a local minimum at (3, −22).

"Local" is doing work there. The cubic in Figure 1 carries on upward forever on the right and downward forever on the left, so (−1, 10) is not the highest point on the graph; there is no highest point. It is only the highest point in its neighbourhood.

On a restricted domain the end points join the competition, as Figure 2 shows. Take the same cubic on −3 ≤ x ≤ 6. Now check four candidates: the two turning points and the two end points.

The greatest value on this domain is 59, at the end point (6, 59), not at the local maximum. The least value is −22, reached twice. So the range of f on this domain is −22 ≤ f(x) ≤ 59, the same method as the range questions in 2.2.

On a restricted domain, the greatest and least values can be at the end points. Evaluate f at every end point as well as every turning point, then compare.

5Symmetry and the vertex

A graph has line symmetry in a vertical line x = h if folding along that line puts one half exactly onto the other. Every parabola has one, its axis of symmetry, and the turning point on it is the vertex. Figure 3(a) shows y = −2x² + 8x − 3. The formula booklet gives the axis of a quadratic ax² + bx + c as x = −b ÷ 2a:

The symmetry gives you points for free: since (0.5, 0.5) is on the curve, so is its mirror image (3.5, 0.5), the same distance on the other side of x = 2. It also means the two x-intercepts of a parabola sit equally either side of its axis, so the axis is halfway between them. For f(x) = (x − 1)(x + 3), the zeros are 1 and −3, so the axis is x = −1 without any formula.

The special case of line symmetry in the y-axis (x = 0) is worth knowing by its algebra: it happens exactly when f(−x) = f(x) for every x. Even powers do this: x², x⁴, and 10 ÷ (x² + 1).

A graph has rotational symmetry about the origin if turning it through half a turn (180°) about (0, 0) leaves it unchanged. Figure 3(b) shows y = x³ − 4x: the point (1, −3) turns onto (−1, 3), and every point on one side matches one on the other, with the origin exactly halfway. The algebra is f(−x) = −f(x). Odd powers do this: x, x³, and 4x ÷ (x² + 1).

When a question asks you to describe a symmetry, name the line (x = 2) or the centre (the origin) precisely.

6Asymptotes

An asymptote is a straight line that a graph gets closer and closer to as it heads off towards infinity. At standard level you need two kinds, and the guide expects you to find them using graphing technology. Figure 4 shows both for y = (2x + 1) ÷ (x − 3).

A vertical asymptote, x = a, is where the function is undefined because it would divide by zero, and nearby the outputs grow without limit, up or down. Here the denominator x − 3 is zero at x = 3. The GDC shows the curve shooting off the top and bottom of the screen either side of x = 3, and the table shows why:

A horizontal asymptote, y = b, is the value the outputs settle towards as x becomes very large, positive or negative. Scroll the table far out:

The outputs settle on 2, so the horizontal asymptote is y = 2. On Paper 1 you can see it without a table: for very large x the +1 and the −3 hardly matter, so (2x + 1) ÷ (x − 3) behaves like 2x ÷ x = 2. The other features follow in the usual way: the y-intercept is (0, 1 ÷ (−3)) = (0, −1/3), and the x-intercept is where the top is zero, 2x + 1 = 0, at (−0.5, 0).

Always write an asymptote as the equation of a line: x = 3 and y = 2, never just "3" and "2". On a sketch, draw it dashed and label it with that equation.

A horizontal asymptote is about what happens far away, not close up, so a graph can cross its horizontal asymptote. Figure 5 shows y = 4x ÷ (x² + 1), whose horizontal asymptote is y = 0: it passes straight through the origin, turns at (1, 2) and (−1, −2), and only then settles towards y = 0 on both sides. A graph can never cross a vertical asymptote, because the function has no value there.

Two warnings about technology. A GDC in connected mode sometimes draws a steep near-vertical line joining the two branches at x = 3. That line is not part of the graph; the table proves it. And a GDC window that stops at x = 10 can make an approach to an asymptote look like a curve that is still falling. Check the table at large x before you claim an asymptote.

7Where two graphs meet

Two graphs meet where they have the same y-value at the same x. So the x-coordinates of the points of intersection of y = f(x) and y = g(x) are exactly the solutions of f(x) = g(x). That turns an equation into a picture, and a picture is something the GDC can read.

That matters because most equations cannot be solved by algebra. There is no algebraic method on the syllabus for 2ˣ = x + 3. The GDC solves it in seconds, two ways, as Figure 6 shows.

Method 1: intersect. Graph y = 2ˣ and y = x + 3 and use the intersect tool at each crossing. The graphs meet at (−2.86, 0.137) and (2.44, 5.44), so the solutions are x = −2.86 and x = 2.44.

Method 2: zeros of the difference. Rearrange to 2ˣ − x − 3 = 0, graph y = 2ˣ − x − 3, and use the zero tool twice. You get the same x-values. This is the method to reach for when one of the two graphs is awkward to enter, or when you want to be sure you have found every solution: count the zeros in a wide window.

Two checks before you write the answer. Have you found them all? Zoom out; an exponential and a line, or two curves, can meet more than once, and the second meeting is often off the default screen. Does the question want x-values or points? "Solve" wants x = …; "find the coordinates" wants (x, y).

Intersections have a real-world meaning whenever the two functions do. In economics, a market equilibrium is where a demand curve meets a supply curve: the price at which the quantity buyers want equals the quantity sellers offer. In 2.3, the break-even points were where revenue met cost. In each case, name the meaning in your answer, with units.