x-axis

(noun)

The axis on a graph that is usually drawn from left to right, with values increasing to the right.

Related Terms

Examples of x-axis in the following topics:

  • Reflections

    • Reflections are a type of transformation that move an entire curve such that its mirror image lies on the other side of the $x$ or $y$-axis.
    • The reflection of a function can be performed along the $x$-axis, the $y$-axis, or any line.  
    • A vertical reflection is a reflection across the $x$-axis, given by the equation:
    • The result is that the curve becomes flipped over the $x$-axis.  
    • Calculate the reflection of a function over the $x$-axis, $y$-axis, or the line $y=x$

  • Zeroes of Linear Functions

    • Graphically, where the line crosses the $x$-axis, is called a zero, or root.  
    • If there is a horizontal line through any point on the $y$-axis, other than at zero, there are no zeros, since the line will never cross the $x$-axis.  
    • If the horizontal line overlaps the $x$-axis, (goes through the $y$-axis at zero) then there are infinitely many zeros, since the line intersects the $x$-axis multiple times.  
    • Because the $x$-intercept (zero) is a point at which the function crosses the $x$-axis, it will have the value $(x,0)$, where $x$ is the zero.
    • To find the zero of a linear function, simply find the point where the line crosses the $x$-axis.

  • Continuous Probability Functions

    • We define the function f (x) so that the area between it and the x-axis is equal to a probability.
    • The area between f (x) = 1 20 where 0 ≤ x ≤ 20 and the x-axis is the area of a rectangle with base = 20 and height = 1 20.
    • This particular function, where we have restricted x so that the area between the function and the x-axis is 1, is an example of a continuous probability density function.
    • Suppose we want to find the area between f (x) = 1 20 and the x-axis where 0 < x < 2 .
    • Suppose we want to find the area between f (x) = 1/20 and the x-axis where 4 < x < 15 .

  • The Cartesian System

    • The horizontal axis is known as the $x$-axis, and the vertical axis is known as the $y$-axis.
    • Each point can be represented by an ordered pair $(x,y) $, where the $x$-coordinate is the point's distance from the $y$-axis and the $y$-coordinate is the distance from the $x$-axis.
    • On the $x$-axis, numbers increase toward the right and decrease toward the left; on the $y$-axis, numbers increase going upward and decrease going downward.
    • The non-integer coordinates $(-1.5,-2.5)$ lie between -1 and -2 on the $x$-axis and between -2 and -3 on the $y$-axis.
    • The revenue is plotted on the $y$-axis, and the number of cars washed is plotted on the $x$-axis.

  • Basics of Graphing Exponential Functions

    • As $x$ takes on smaller and smaller values the curve gets closer and closer to the $x$-axis.
    • As $x$ takes on smaller and smaller values the curve gets closer and closer to the $x$ -axis.
    • As you can see in the graph below, the graph of $y=\frac{1}{2}^x$ is symmetric to that of $y=2^x$ over the $y$-axis.
    • The function $y=b^x$ has the $x$-axis as a horizontal asymptote because the curve will always approach the $x$-axis as $x$ approaches either positive or negative infinity, but will never cross the axis as it will never be equal to zero.
    • The $x$-axis is a horizontal asymptote of the function.

  • Area of a Surface of Revolution

    • If the curve is described by the function $y = f(x) (a≤x≤b)$, the area $A_y$ is given by the integral $A_x = 2\pi\int_a^bf(x)\sqrt{1+\left(f'(x)\right)^2} \, dx$ for revolution around the $x$-axis.
    • Examples of surfaces generated by a straight line are cylindrical and conical surfaces when the line is co-planar with the axis, as well as hyperboloids of one sheet when the line is skew to the axis.
    • If the curve is described by the parametric functions $x(t)$, $y(t)$, with $t$ ranging over some interval $[a,b]$ and the axis of revolution the $y$-axis, then the area $A_y$ is given by the integral:
    • Likewise, when the axis of rotation is the $x$-axis, and provided that $y(t)$ is never negative, the area is given by:
    • A portion of the curve $x=2+\cos z$ rotated around the $z$-axis (vertical in the figure).

  • Double Integrals Over General Regions

    • the projection of $D$ onto either the $x$-axis or the $y$-axis is bounded by the two values, $a$ and $b$.
    • $x$-axis: If the domain $D$ is normal with respect to the $x$-axis, and $f:D \to R$ is a continuous function, then $\alpha(x)$  and $\beta(x)$ (defined on the interval $[a, b]$) are the two functions that determine $D$.
    • $\displaystyle{\iint_D f(x,y)\ dx\, dy = \int_a^b dx \int_{ \alpha (x)}^{ \beta (x)} f(x,y)\, dy}$
    • $y$-axis: If $D$ is normal with respect to the $y$-axis and $f:D \to R$ is a continuous function, then $\alpha(y)$ and $\beta(y)$ (defined on the interval $[a, b]$) are the two functions that determine $D$.
    • In this case the two functions are $\alpha (x) = x^2$ and $\beta (x) = 1$, while the interval is given by the intersections of the functions with $x=0$, so the interval is $[a,b] = [0,1]$ (normality has been chosen with respect to the $x$-axis for a better visual understanding).

  • Cylindrical Shells

    • The idea is that a "representative rectangle" (used in the most basic forms of integration, such as $\int x \,dx$) can be rotated about the axis of revolution, thus generating a hollow cylinder with infinitesimal volume.
    • The volume of the solid formed by rotating the area between the curves of $f(x)$ and $g(x)$ and the lines $x=a$ and $x=b$ about the $y$-axis is given by:
    • $\displaystyle{V = 2\pi \int_a^b x \left | f(x) - g(x) \right | \,dx}$
    • The volume of solid formed by rotating the area between the curves of $f(y)$ and and the lines $y=a$ and $y=b$ about the $x$-axis is given by:
    • Each segment located at $x$, between $f(x)$and the $x$-axis, gives a cylindrical shell after revolution around the vertical axis.

  • Volumes of Revolution

    • The volume of the solid formed by rotating the area between the curves of $f(x)$ and $g(x)$ and the lines $x=a$ and $x=b$ about the $x$-axis is given by:
    • If $g(x) = 0$ (e.g. revolving an area between curve and $x$-axis), this reduces to:
    • The volume of the solid formed by rotating the area between the curves of $f(x)$and $g(x)$ and the lines $x=a$ and $x=b$ about the $y$-axis is given by:
    • If $g(x)=0$ (e.g. revolving an area between curve and $x$-axis), this reduces to:
    • The integration (along the $x$-axis) is perpendicular to the axis of revolution ($y$-axis).

  • Standard Equations of Hyperbolas

    • A standard equation for a hyperbola can be written as $x^2/a^2 - y^2/b^2 = 1$.
    • Consistent with the symmetry of the hyperbola, if the transverse axis is aligned with the x-axis, the slopes of the asymptotes are equal in magnitude but opposite in sign, ±b⁄a, where b=a×tan(θ) and where θ is the angle between the transverse axis and either asymptote.
    • A conjugate axis of length 2b, corresponding to the minor axis of an ellipse, is sometimes drawn on the non-transverse principal axis; its endpoints ±b lie on the minor axis at the height of the asymptotes over/under the hyperbola's vertices.
    • If the transverse axis of any hyperbola is aligned with the x-axis of a Cartesian coordinate system and is centered on the origin, the equation of the hyperbola can be written as:
    • The two thick black lines parallel to the conjugate axis (thus, perpendicular to the transverse axis) are the two directrices, D1 and D2.