Intersecting with the x-axis / intersecting the x-axis

Which is correct?: "The function intersects with the x-axis" or "The function intersects the x-axis"

Is the verb 'to intersect' in the mathematical sense accompanied by the preposition 'with'?

The second is correct. Geometrically, lines "intersect one another". It is incorrect to include "with", which would render the verb "intersect" intransitive. The geometrical meaning is OED - sense 1b in the OED:

1b. Geometry. Of a line or surface: To pass through or across (a line or surface), so as to lie on both sides of it with one point (or line) in common: = cut v. 15b.

1646 Sir T. Browne Pseudodoxia Epidemica vi. v. 292 Being in the Æquator it would intersect their Horizon, and be halfe above and halfe beneath it.

1700 Moxon's Mech. Exercises: Bricklayers-wks. 30 Where these two Arches Intersect, or cut each other, there is the Center.

1840 D. Lardner Treat. Geom. x. 125 If two chords intersect each other in a circle, the rectangle under the segments of the one will be equal to the rectangle under the segments of the other.

1858 O. W. Holmes Autocrat of Breakfast-table xii. 330 Keep any line of knowledge ten years and some other line will intersect it. 1873 B. Williamson Elem. Treat. Differential Calculus (ed. 2) xiii. §190 Every [straight] line must intersect a curve of an odd degree in at least one real point.

There is an intransitive form of "intersect", but it does not involve the use of "with". It is used where the intersecting parties are collectively the subject of the verb. It is sense 2a.

2a. intransitive (for reflexive). To cross or cut each another: chiefly Geometry of lines or surfaces. 1755 in Johnson: quoting Wiseman in error: see quot. 1676 at interject v. 2a.

1849 G. Grote Hist. Greece VI. ii. xlvii. 27 Straight streets intersecting at right angles.

1869 J. Tyndall Notes 9 Lect. on Light 24 The rays from a luminous point placed beyond the focus intersect at the opposite side of the lens.

1873 B. Williamson Elem. Treat. Differential Calculus (ed. 2) xiv. §204 The Lemniscate whose equation is (x2 + y2)2 = a (x2 − y2)..[has] two branches intersecting at the origin.

It would seem that the use of "intersect with" is an incorrect form.

• It looks like each blockquote includes a number of example sentences; I'd suggest formatting them to appear on separate lines to make it easier to read. – V2Blast May 10 '19 at 7:19
• @V2Blast I used always to do that, but it is really quite a lot of bother. I think if people are interested enough they will do that for themselves. – WS2 May 11 '19 at 15:01
• The quote is nigh-unreadable with everyone mashed together in one line. It'd be easy to fix in about 2 minutes, but I don't know which citation is associated with which quote. – V2Blast May 11 '19 at 18:15
• @V2Blast Every quotation begins with a year date. So where you see a year date -start a new line if you must. – WS2 May 11 '19 at 19:21
• Thanks, it's much clearer now. – V2Blast May 11 '19 at 19:55

While one might interact with something, one intersects something. From Merriam-Webster:

intersect verb

in·​ter·​sect | \ ˌin-tər-ˈsekt \

intersected; intersecting; intersects

Definition of intersect

transitive verb

: to pierce or divide by passing through or across : CROSS

// a comet intersecting earth's orbit

// one line intersects another

intransitive verb

1 : to meet and cross at a point

// lines intersecting at right angles

2 : to share a common area : OVERLAP

//where morality and self-interest intersect