The slope of a line in the plane is defined as the rise over the run, m = Δy/Δx.

In mathematics, the slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline. Slope is not defined for vertical lines.

Slope is normally described by the ratio of the "rise" divided by the "run" between two points on a line. The line may be practical - a set by a road surveyor - or in a diagram that models a road or a roof either as a description or as a plan.

For relatively short distances - where the earth's curvature may be neglected, the rise of a road between two points is the difference between the altitude of at those two points, say y₁ and y₂, or in other words,

the rise is (y₂ − y₁) = Δy.

In this example, the run is the difference in distance from a fixed point a measured along a level, horizontal line, or in other words,

the run is (x₂ − x₁) = Δx.

Here the slope of the road between the two points is simply described as the ratio of the altitude change to the horizontal distance between any two points on the line. In mathematical language,

the slope m of the line is

Failed to parse (Missing texvc executable; please see math/README to configure.): m=\frac{y_2-y_1}{x_2-x_1}.

The concept of slope applies directly to grades or gradients in geography and civil engineering. Through trigonometry, the grade m of a road is related to its angle of incline θ by

Failed to parse (Missing texvc executable; please see math/README to configure.): m = \tan \theta.\!

As a generalization of this practical description, the mathematics of differential calculus defines the slope of a curve at a point as the slope of the tangent line at that point. When the curve given by a series of points in a diagram or in a list of the coordinates of points, the slope may be calculated not at a point but between any two given points. When the curve is given as a continuous function, perhaps as an algebraic formula, then the differential calculus provides rules giving a formula for the slope of the cure at any point in the middle of the curve.

This generalization of the concept of slope allows very complex constructions to be planned and built that go well beyond static structures that are either horizontals or verticals, but can change in time, move in curves, and change depending on the rate of change of other factors. Thereby, the simple idea of slope becomes one of the main basis of the modern world in terms of both technology and the built environment.

Definition[link]

The slope of a line in the plane containing the x and y axes is generally represented by the letter m, and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line. This is described by the following equation:

Failed to parse (Missing texvc executable; please see math/README to configure.): m = \frac{\Delta y}{\Delta x} = \frac{\text{rise}}{\text{run}}.

(The delta symbol, "Δ", is commonly used in mathematics to mean "difference" or "change".)

Given two points (x₁,y₁) and (x₂,y₂), the change in x from one to the other is x₂ − x₁ (run), while the change in y is y₂ − y₁ (rise). Substituting both quantities into the above equation generates the formula:

Failed to parse (Missing texvc executable; please see math/README to configure.): m = \frac{y_2 - y_1}{x_2 - x_1}.

The formula fails for a vertical line, parallel to the y axis (see Division by zero), where the slope can be taken as infinite, so the slope of a vertical line is considered undefined.

Examples[link]

Suppose a line runs through two points: P = (1, 2) and Q = (13, 8). By dividing the difference in y-coordinates by the difference in x-coordinates, one can obtain the slope of the line:

Failed to parse (Missing texvc executable; please see math/README to configure.): m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{8 - 2}{13 - 1} = \frac{6}{12} = \frac{1}{2}.

As another example, consider a line which runs through the points (4, 15) and (3, 21). Then, the slope of the line is

Failed to parse (Missing texvc executable; please see math/README to configure.): m = \frac{ 21 - 15}{3 - 4} = \frac{6}{-1} = -6.

Geometry[link]

The larger the absolute value of a slope, the steeper the line. A horizontal line has slope 0, a 45° rising line has a slope of +1, and a 45° falling line has a slope of −1. A vertical line's slope is undefined.

The angle θ a line makes with the positive x axis is closely related to the slope m via the tangent function:

Failed to parse (Missing texvc executable; please see math/README to configure.): m = \tan\,\theta

and

Failed to parse (Missing texvc executable; please see math/README to configure.): \theta = \arctan\,m

(see trigonometry).

Two lines are parallel if and only if their slopes are equal and they are not coincident or if they both are vertical and therefore have undefined slopes. Two lines are perpendicular if the product of their slopes is −1 or one has a slope of 0 (a horizontal line) and the other has an undefined slope (a vertical line). Also, another way to determine a perpendicular line is to find the slope of one line and then to get its reciprocal and then reversing its positive or negative sign (e.g. a line perpendicular to a line of slope  −2 is +1/2).

Slope of a road or railway[link]

Main articles: Grade (slope), Grade separation

There are two common ways to describe how steep a road or railroad is. One is by the angle in degrees, and the other is by the slope in a percentage. See also mountain railway and rack railway. The formulae for converting a slope as a percentage into an angle in degrees and vice versa are:

Failed to parse (Missing texvc executable; please see math/README to configure.): \text{angle} = \arctan \frac{\text{slope}}{100} ,

and

Failed to parse (Missing texvc executable; please see math/README to configure.): \mbox{slope} = 100 \tan( \mbox{angle}),\,

where angle is in degrees and the trigonometric functions operate in degrees. For example, a 100% or 1000‰ slope is 45°.

A third way is to give one unit of rise in say 10, 20, 50 or 100 horizontal units, e.g. 1:10. 1:20, 1:50 or 1:100 (etc.).

• 

  Slope warning sign in the Netherlands

• 

  Slope warning sign in Poland

• 

  A 1371-meter distance of a railroad with a 20‰ slope. Czech Republic

• 

  Steam-age railway gradient post indicating a slope in both directions at Meols railway station, United Kingdom

Algebra[link]

If y is a linear function of x, then the coefficient of x is the slope of the line created by plotting the function. Therefore, if the equation of the line is given in the form

Failed to parse (Missing texvc executable; please see math/README to configure.): y = mx + b \,

then m is the slope. This form of a line's equation is called the slope-intercept form, because b can be interpreted as the y-intercept of the line, the y-coordinate where the line intersects the y-axis.

If the slope m of a line and a point (x₁,y₁) on the line are both known, then the equation of the line can be found using the point-slope formula:

Failed to parse (Missing texvc executable; please see math/README to configure.): y - y_1 = m(x - x_1).\!

For example, consider a line running through the points (2,8) and (3,20). This line has a slope, m, of

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac {(20 - 8)}{(3 - 2)} \; = 12. \,

One can then write the line's equation, in point-slope form:

Failed to parse (Missing texvc executable; please see math/README to configure.): y - 8 = 12(x - 2) = 12x - 24 \,

or:

Failed to parse (Missing texvc executable; please see math/README to configure.): y = 12x - 16. \,

The slope of the line defined by the linear equation

Failed to parse (Missing texvc executable; please see math/README to configure.): ax + by +c = 0 \,

is: Failed to parse (Missing texvc executable; please see math/README to configure.): -\frac {a}{b} \; .

Calculus[link]

At each point, the derivative is the slope of a line that is tangent to the curve. The line is always tangent to the blue curve; its slope is the derivative. Note derivative is positive where green, negative where red, and zero where black

The concept of a slope is central to differential calculus. For non-linear functions, the rate of change varies along the curve. The derivative of the function at a point is the slope of the line tangent to the curve at the point, and is thus equal to the rate of change of the function at that point.

If we let Δx and Δy be the distances (along the x and y axes, respectively) between two points on a curve, then the slope given by the above definition,

Failed to parse (Missing texvc executable; please see math/README to configure.): m = \frac{\Delta y}{\Delta x}

,

is the slope of a secant line to the curve. For a line, the secant between any two points is the line itself, but this is not the case for any other type of curve.

For example, the slope of the secant intersecting y = x² at (0,0) and (3,9) is 3. (The slope of the tangent at x = ³⁄₂ is also 3—a consequence of the mean value theorem.)

By moving the two points closer together so that Δy and Δx decrease, the secant line more closely approximates a tangent line to the curve, and as such the slope of the secant approaches that of the tangent. Using differential calculus, we can determine the limit, or the value that Δy/Δx approaches as Δy and Δx get closer to zero; it follows that this limit is the exact slope of the tangent. If y is dependent on x, then it is sufficient to take the limit where only Δx approaches zero. Therefore, the slope of the tangent is the limit of Δy/Δx as Δx approaches zero, or dy/dx. We call this limit the derivative.

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{dy}{dx} = \lim_{\Delta x \to 0}\frac{\Delta y}{\Delta x}

See also[link]

• The gradient is a generalization of the concept of slope for functions of more than one variable.
• Slope definitions
• Euclidean distance
• Theil–Sen estimator, a line with the median slope among a set of sample points

External links[link]

Look up slope in Wiktionary, the free dictionary.

The Dø
The Dø performing at the Paléo Festival in 2008.
Background information
Origin	France, Helsinki, Finland
Genres	Folk rock, indie rock
Years active	2007–present
Labels	Get DOwn! Cinq7 (Fr) Six Degrees (US) AK Music Video (Tu) Andante Vs Massala (Pt) Ministry of Sound (De) Rockhopper Music (Fi) Border Music (Se) Shock Records (Au) Bang Music (Be)
Website	Official website
Members
Dan Levy Olivia Bouyssou Merilahti

The Dø is a French/Finnish indie pop band founded in Paris in 2005. The band is Olivia Merilahti (singer and musician) and Dan Levy (multi-instrumentalist). The duo has been backed on stage by three different drummers (Jérémie Pontier 2007–2008, José Joyette 2008–2009, Pierre Belleville since 2009). Their first studio album A Mouthful topped the French charts in 2008, making them the first French act singing in English to reach that position.

Biography[link]

Formation and first years (2005–2007)[link]

Olivia Merilahti and Dan Levy met in 2005 while recording the music for a French movie Empire of the Wolves directed by Chris Nahon and produced by Gaumont.

They released their first piece under the name The Dø a few months later: a 3-track EP including The Bridge is Broken for a contemporary dance ballet titled Scène d’amour by Finnish choreographer Juha-Pekka Marsalo. They kept on working for cinema (The Passenger, awarded at the Festival d'Angers and the Festival d'Aubagne, Wild Camp, and Darling), dance (Prologue, Perle and Cinderella by Juha-Pekka Marsalo, as well as poetry lectures by Carolyn Carlson) and theatre (Laure by Colette Peignot).

They used their free time to write the songs that would feature on their first album, A Mouthful. The band's myspace page was opened in 2007. The first four songs released online (The Bridge is Broken, At Last, On My Shoulders and Playground Hustle) quickly created a buzz that brought them to play a series of sold-out concerts in Paris. In December 2007, they opened the festival Les Transmusicales de Rennes.

Meanwhile, they signed on the indie label Cinq/7, and the song On My Shoulders was their first single, also used in an Oxford-notepads commercial.

The band's name is derived from the first note of the solfège scale, and is pronounced as the English word "dough", with a long "o" sound. It is written with the letter Ø (slashed o), and the "D" often in lower-case to resemble a half note.

[edit] A Mouthful (2008–2010)

A Mouthful was released in France in January 2008. The rest of Europe and Australia released it later that year . It was very well received by critics and the public, ranking n°1 in the French charts on the first week. It was acclaimed as an eclectic record, drawing inspiration from various genres as pop, folk, rock or hip-hop, while keeping an integrity and an original approach.

The first single from the album was On my shoulders, followed by At last and Stay (Just a little bit more). The music video for At last was directed by Belgian choreographer Wim Vandekeybus.

The band went on tour in Europe and Australia for over 200 shows. They spent summer 2008 playing at most of the biggest festivals in Europe (Eurockéennes de Belfort, Paleo, Provinssirock, Vieilles Charrues, Roskilde, Main Square Festival, Pukkelpop...), flew to Australia for the V Festival in spring 2009, and spent two weeks in America, playing from Mexico to Canada, NYC to Los Angeles, in September 2009.

A Mouthful was officially released in the US in April 2010 on Six Degrees Records, including 3 bonus tracks.

[edit] Both Ways Open Jaws (2011)

In December 2010, the band released a video for the first promotional single 'Dust It Off'. Their second album entitled Both Ways Open Jaws was released in some parts of Europe on Monday, 7 March 2011, and in France on Wednesday, 9 March 2011. Beginning late October and ending in mid-November, the album was released worldwide.

Discography[link]

Albums[link]

• 2005: Empire of the Wolves / Soundtrack^[1]
• A Mouthful (2008)^[2]
• A Mouthful [U.S. Release] (2010)^[3]
• Dust It Off - EP (2010)^[4]
• Both Ways Open Jaws (2011)
• Both Ways Open Jaws Extended (2011)

Singles[link]

• "On My Shoulders" (2007)
• "At Last!" (2008)

References[link]

External links[link]

Wikimedia Commons has media related to: The Dø

• Official website
• The Dø 'A Mouthful' Album Review
• Six Degrees Records

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (May 2008)

John Edward Hawkins (November 15, 1969 – May 1, 2006), better known as H.A.W.K. or Big Hawk was an American rapper from Houston, Texas and a founding member of the late DJ Screw's rap group the Screwed Up Click.

Biography[link]

Born John Edward Hawkins in Houston, Texas on November 15, 1969. He grew up on the dead end block of MLK with his younger brother, Patrick "Fat Pat" Hawkins and two sisters. He started rapping in 1992 when Pat took him to DJ Screw’s house, an upcoming mixtape producer and DJ in the southern area of Houston.^[4] In April 2006 Hawk married his longtime girlfriend and mother of his two children, Meshah Henderson.^[4]

Career[link]

In 1994 Hawk, Fat Pat, DJ Screw and some of their friends, Lil' KeKe and Koldjack, collaborated to form the group D.E.A. and Dead End Records (named for the dead end block of Martin Luther King Blvd where Hawk grew up in Houston). In late 1995 D.E.A. released an original independent album entitled, “Screwed For Life” featuring Lil' Keke, Big Pokey and the rest of the S.U.C. After the deaths of Hawk’s brother (Fat Pat) and DJ Screw and the incarceration of many others SUC members, Hawk’s ultimate mission and goal became keeping the “Ghetto Dream” alive. In 1999 Hawk participated in a Southside Playaz compilation album titled “You Got Us Fuxxed Up,” with Mike D, Claydoe and other members of the S.U.C. Hawk released his solo album, "Under Hawk's Wings" on Dead End Records in 2000 and had recorded with Lil' Flip, Lil' Keke, Big Moe, Z-Ro and was featured on the Lil' Troy hit "Wanna Be A Baller."

Big Hawk appeared on numerous mixtapes from artists in the Houston underground rap scene. After Screw's death in 2000, he was the main person responsible for keeping the S.U.C. together,^{[citation needed]} so much so that his nickname was the Five Star General of the Screwed Up Click. He collaborated with Lil' O on the smash hit, “Back Back” Next, he partnered with Game Face Records in 2002 and released his first album under his label Ghetto Dreams Ent., self titled, “HAWK.” With the hit song, “U Already Know,” the album charted the Billboard’s list of top R&B and Rap Albums at no. 45. It was at Game Face Records where Hawk met and clicked with a young up and coming group called the GritBoyz and an artist named Starchy Archy. Hawk and Ghetto Dreams Ent. teamed up with Presidential Records in 2003 to release “A Bad Azz Mix Tape Vol. II.” A year later he and Lil' Keke teamed up to release “Wreckin 2K4” with Presidential Records as well. It was around this time when the original S.U.C. members, along with M.J. (of Southside Studios) started collaborating on a group/compilation album titled “The Takeover.” Trae tha Truth made an album dedicated to BIG HAWK called Life Goes On.

Hawk was featured on the popular single "Swang" by Trae that included a tribute to Fat Pat in 2005. In 2006, Hawk, along with Clint Dempsey of the U.S. National Soccer team recorded a song for Nike's "Joga Bonito" World Cup soccer promotion called "Don't Tread". He was part of Pepsi's Yahoo! "Houston Mic Pass" with Lil' Flip, Chamillionaire, Trae, Chingo Bling, Bun B, Short Dawg, Magno and Mike D. Since his death, Hawk has been featured on several tracks including Lil O's "I Do" single, "Down In Texas" off of the 2008 A.B.N. album It Is What It Is, and Big Unk's "I'ma Beast" to name a few.

On November 11, 2011 two posthumous singles were released called "Somebody Who Loves You" and "Praise God."

Death[link]

Hawkins, 36, was shot to death Monday, May 1, 2006, outside of a home in south Houston.^[5] He was pronounced dead at the scene. Police say the rapper had gone to visit a friend's house at night and was walking from his car when he was shot several times. There is no known suspect.^[5]

Shortly after his death Bun B, a fellow Houston rap artist, spoke about Hawkins in an interview with the Houston Chronicle saying, "There were no chinks in this man's moral armor, because he was a rapper there will be people who will take his death the wrong way, but he was a peaceful guy and a family man who had no beefs with anybody."^[6]

Discography[link]

Albums[link]

Mixtapes[link]

• 2004: A Bad Azz Mix Tape Vol. II.^[7]
• 2006: Since the Gray Tapes, Vol. 4^[7]
• 2007: Terrible Texas Dub K Six^[7]

See also[link]

• Houston hip hop

References[link]

External links[link]

• Hawk's MySpace