When y is viewed as a function of x, the cycloid is differentiable everywhere except at the cusps where it hits the x-axis, with the derivative tending toward or as one approaches a cusp.

Although the Engel curve remains upward sloping in both cases, it bends toward the y-axis for necessities and towards the x-axis for luxury goods.

R < sub > z </ sub >, for instance, would rotate toward the y-axis a vector aligned with the x-axis.

Differing from the traditional " left / right " distinction and other political taxonomies, the Nolan Chart in its original form has two dimensions, with a horizontal x-axis labeled " economic freedom " and a vertical y-axis labeled " personal freedom ".

In general, if y = f ( x ), then it can be transformed into y = af ( b ( x − k )) + h. In the new transformed function, a is the factor that vertically stretches the function if it is greater than 1 or vertically compresses the function if it is less than 1, and for negative a values, the function is reflected in the x-axis.

The tension at r is parallel to the curve at r and pulls the section to the right, so it may be written Tu =( Tcos φ, Tsin φ ), where T is the magnitude of the force and φ is the is the angle between the curve at r and the x-axis ( see tangential angle ).

Here, i is the complex number whose square is the real number − 1 and is identified with the point with coordinates ( 0, 1 ), so it is not the unit vector in the direction of the x-axis ( this confusion is just an unfortunate historical accident ).

If A is nonzero, then the x-intercept, that is, the x-coordinate of the point where the graph crosses the x-axis ( where, y is zero ), is − C / A.

This line intersects the y-axis at the point ( 0 ,- y ) = ( 0 ,-a x² ), and the x-axis at the point ( x / 2, 0 ).

In the context of the standard unit circle with radius 1, where a triangle is formed by a ray originating at the origin and making some angle with the x-axis, the sine of the angle gives the length of the y-component ( rise ) of the triangle, the cosine gives the length of the x-component ( run ), and the tangent function gives the slope ( y-component divided by the x-component ).

The unit vectors appropriate to cylindrical symmetry are: ( also designated or ), the distance from the axis of symmetry ;, the angle measured counterclockwise from the positive x-axis ; and.

In general, something that is horizontal can be drawn from left to right ( or right to left ), such as the x-axis in the Cartesian coordinate system.

For a continuous function of several variables, the meaning of the definition is the same, except for the fact that the continuous path to be considered cannot be the whole graph of the given function ( which is a hypersurface in this case ), but can be every intersection of the graph itself with a hyperplane ( in the case of functions of two variables, a plane ) parallel to a fixed x-axis and to the y-axis.

* An horocycle ( a curve which is like a circle with an infinite radius ) is modeled by a circle tangent to the x-axis ( but excluding the point of intersection ), or by a line parallel to the x-axis.

The coefficient of elasticity of any linear supply curve that cuts the y-axis is greater than 1 ( elastic ), and the coefficient of elasticity of any linear supply curve that cuts the x-axis is less than 1 ( inelastic ).

Generally chromatographic data is presented as a graph of detector response ( y-axis ) against retention time ( x-axis ), which is called a chromatogram.

where is the reduced Planck's constant, m is the particle mass, x represents distance measured in the direction of motion of the particle, Ψ is the Schrödinger wave function, V is the potential energy of the particle ( measured relative to any convenient reference level ), E is the energy of the particle that is associated with motion in the x-axis ( measured relative to V ), and M ( x ) is a quantity defined by V ( x )-E which has no accepted name in physics.

* θ is the angle between the projection of the vector onto the xy-plane ( i. e. r ) and the positive x-axis ( 0 ≤ θ < 2π ),

The dotted line in Figure 1 shows the optimal probability that player Y plays ' Stag ' ( in the y-axis ), as a function of the probability that player X plays Stag ( shown in the x-axis ).

In Figure 2 the dotted line shows the optimal probability that player X plays ' Stag ' ( shown in the x-axis ), as a function of the probability that player Y plays Stag ( shown in the y-axis ).

γ ( t ) = ( x ( t ), y ( t )), then the parametric equation of the evolute can be expressed in terms of the radius of curvature R = 1 / k and the tangential angle φ, which is the angle the tangent to the curve makes with a fixed reference axis x-axis.

The coordinates have two foci F < sub > 1 </ sub > and F < sub > 2 </ sub >, which are generally taken to be fixed at (− a, 0 ) and ( a, 0 ), respectively, on the x-axis of a Cartesian coordinate system.

Let Q be the semicircle with diameter on the x-axis that passes through the points ( 1, 0 ) and ( 0, y ), where y > 1.

Most psychrometric charts used in HVAC design only show the dry bulb temperature on the x-axis ( abscissa ), however, it is the operative temperature which is specified on the x-axis of the psychrometric chart illustrated in ANSI / ASHRAE Standard 55 – Thermal Environmental Conditions for Human occupancy.

Diminishing Marginal returns, refers to the point where the MPP curve starts to slope down and travels all the way down to the x-axis and beyond.

Intuitively, the process can be pictured as follows: first shrink the real line to the open interval (- π, π ) on the x-axis ; then bend the ends of this interval upwards ( in positive y-direction ) and move them towards each other, until you get a circle with one point ( the topmost one ) missing.

Thus the " correct " way to view Figure 8 is to imagine the x-axis as pointing towards the observer and thus seeing a concave corner.

Coordinate systems fixed with respect to the rotating axles are defined as having their x-axis unit vectors ( and ) pointing from the origin towards one of the connection points.

The southern cross is tilted so that Gamma Crucis stays in the same position as for the Australian National Flag and that Alpha Crucis is moved along the x-axis towards the hoist by one-sixth of the width of the flag.

But no matter how large x becomes, its reciprocal 1 / x is never 0, so the curve never actually touches the x-axis.

As the name indicate they are parallel to the x-axis.

Vertical asymptotes are vertical lines ( perpendicular to the x-axis ) near which the function grows without bound.

The scheme graphs average annual precipitation ( x-axis ) versus average annual temperature ( y-axis ) to classify biome-types.

: Plotted on the x-axis is the retention time and plotted on the y-axis a signal ( for example obtained by a spectrophotometer, mass spectrometer or a variety of other detectors ) corresponding to the response created by the analytes exiting the system.

Euler proved in 1744 that the catenary is the curve which, when rotated about the x-axis, gives the surface of minimum surface area ( the catenoid ) for the given bounding circles.

which is the length of the line normal to the curve between it and the x-axis.

Also, the geometric centroid of the area under a stretch of catenary is the midpoint of the perpendicular segment connecting the centroid of the curve itself and the x-axis.

The surface of revolution with fixed radii at either end that has minimum surface area is a catenary revolved about the x-axis.

and by shifting the position of the x-axis, β can be taken to be 0.

The x-axis thus chosen is called the directrix of the catenary.

The direction of u < sub > r </ sub > is described by θ, the angle between the x-axis and the unit vector, measured counterclockwise from the x-axis.

) The lines are commonly referred to as the x and y-axes where the x-axis is taken to be horizontal and the y-axis is taken to be vertical.

For a given point P, a line is drawn through P perpendicular to the x-axis to meet it at X and second line is drawn through P perpendicular to the y-axis to meet it at Y.

The z-axis is vertical and the x-axis is highlighted in green.

Fixing or choosing the x-axis determines the y-axis up to direction.

Namely, the y-axis is necessarily the perpendicular to the x-axis through the point marked 0 on the x-axis.

The usual way of orienting the axes, with the positive x-axis pointing right and the positive y-axis pointing up ( and the x-axis being the " first " and the y-axis the " second " axis ) is considered the positive or standard orientation, also called the right-handed orientation.

Placing a somewhat closed right hand on the plane with the thumb pointing up, the fingers point from the x-axis to the y-axis, in a positively oriented coordinate system.