A graph can cross its own asymptote

the crossing

y = 0asymptote

crossed at x = −0.5what happens

crossesverdict

An asymptote describes what happens far away. It makes no promise about the middle.

The figure needs JavaScript. It plots y = (2x + 1)/(x² + 1), which has a horizontal asymptote at y = 0, sits on it at x = −0.5, rises to a maximum of 1.618 at x = 0.618 and falls back towards zero; then y = (2x + 1)/(x − 3), which has a vertical asymptote at x = 3 and a horizontal one at y = 2 and never reaches either. Everything below stands on its own.

What an asymptote actually claims

A horizontal asymptote at y = L says: as x runs off to infinity, the outputs approach L. That is all.

For y = (2x + 1)/(x² + 1): at x = 50 the value is 0.04, at a million it is almost nothing. So y = 0 is the asymptote.
And at x = −0.5 the top is 2(−0.5) + 1 = 0, so the whole fraction is 0. The curve is on the line.
No contradiction: the claim was about far away, and x = −0.5 is not far away.

A vertical asymptote is different, and that is where the "never touches" idea comes from. y = (2x + 1)/(x − 3) has one at x = 3, because at x = 3 the bottom is zero and the function has no value at all. It cannot be crossed, because there is nothing there to cross.

The same function's horizontal asymptote is y = 2, and that one really is never reached:

Try to solve it: (2x + 1)/(x − 3) = 2 gives 2x + 1 = 2x − 6, so 1 = −6. No solution.
It gets close from both sides: at x = 100 the value is 2.072, at x = −100 it is 1.932.

So one of these curves crosses its horizontal asymptote and the other cannot, and you can tell which by trying to solve f(x) = L. That takes one line and settles it.

The features, and what each one is called

Feature	What it means	Worked on y = x² − 4x + 1
y-intercept	Where the graph meets the y-axis. Put x = 0.	1
Zeros, or roots	Where it meets the x-axis. Solve f(x) = 0.	0.27 and 3.73
Vertex	The single turning point of a parabola.	(2, −3)
Axis of symmetry	The vertical line through the vertex.	x = 2
Minimum value	The smallest output. For an upward parabola, the vertex's y.	−3

Zero, root and x-intercept are the same thing with three names. A zero of the function, a root of the equation and an x-intercept of the graph all mean the same place. Questions switch between them deliberately, and a student who thinks they are three different things will go looking for something that is not there.

The vertex sits midway between the roots. Here (0.27 + 3.73)/2 = 2, which is the axis of symmetry. That is a free check on both answers, and it works for every parabola with two real roots.

Intersections

Finding where two curves meet is a calculator job and an accuracy trap.

Graph both, then use the intersection tool. Do not read coordinates off the screen by eye.
Two curves can meet more than once, and the tool finds one intersection at a time: the one nearest where you put the cursor. If a question says "find the points", there is more than one.
Give the accuracy the question asks for, from the tool's output, not from the pixel you were looking at.

On the GDC: finding features, one at a time

Every feature in this sub-topic has its own tool, and each one finds exactly one answer per go. Knowing that is the difference between finding both roots and finding one.

When you may use it. Applications allows a calculator in every paper, and this sub-topic assumes one. Analysis does not allow one in Paper 1, where the same features come from the algebra: completing the square for a vertex, the formula for roots.

TI-Nspire CX II

ctrl doc → Add Graphs, then (2x+1)/(x x² +1)
menu → Analyze Graph → Maximum, then click a left bound and a right bound either side of the hill: (0.618, 1.618)
Zero with bounds either side of x = −0.5: -0.5, which is on the asymptote
For an intersection, graph the second function as f2(x), then Analyze Graph → Intersection, bounding one crossing at a time

Casio fx-CG50

MENU → Graph, enter Y1=(2X+1)÷(X²+1), then F6 DRAW
SHIFT F5 G-SOLVE → F2 MAX: (0.618, 1.618)
ROOT for the crossing at -0.5
With two functions entered, G-SOLVE → ISCT, and the left and right arrows step between the intersections

The mark people lose. Finding one root and stopping. Every one of these tools answers once per go, so a question worth two marks for "the zeros" wants you to run it twice. The other one is accuracy: reading 1.6 off the screen when the tool says 1.618 and the question asked for three significant figures. Use the tool's number, not the picture's. And on the Casio, G-SOLVE is SHIFT F5, not F5.

Your turn

1. For y = x² − 4x + 1, what is the y-coordinate of the vertex?

2. For the same parabola, give the larger root to 2 decimal places.

3. For y = (2x + 1)/(x − 3), what is the equation of the vertical asymptote? Type just the x value.

4. For y = (2x + 1)/(x² + 1), give the maximum value to 3 decimal places.

5. A student says a graph can never meet its asymptote. What is the accurate version?

Question 5. A student says a graph can never meet its asymptote. What is the accurate version? It is true for vertical asymptotes and can be false for horizontal ones. It is true for horizontal asymptotes and can be false for vertical ones. It is always true; the crossing on this page is a rounding effect. It is never true for either kind.

Where the marks go

1 markEach feature found, to the stated accuracy.

1 markEach feature named correctly, where the question asks for a description rather than a number.

1 markAn asymptote given as an equation: x = 3, not 3.

That last one is the cheapest mark lost in this sub-topic. An asymptote is a line, so its answer is an equation. "3" is a number and does not name a line.