y-axis

(noun)

The axis on a graph that is usually drawn from bottom to top, with values increasing farther up.

Related Terms

Examples of y-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 horizontal reflection is a reflection across the $y$-axis, given by the equation:
    • The result is that the curve becomes flipped over the $y$-axis.  
    • Calculate the reflection of a function over the $x$-axis, $y$-axis, or the line $y=x$

  • 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 you connect the points, you will notice a smooth curve that crosses the $y$-axis at the point $(0,1)$ and is increasing as $x$ takes on larger and larger values.
    • 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.
    • That is, if the plane were folded over the $y$-axis, the two curves would lie on each other.
    • 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 graph of this function crosses the $y$-axis at $(0,1)$ and increases as $x$ approaches infinity.

  • Graphing Equations

    • For an equation with two variables, $x$ and $y$, we need a graph with two axes: an $x$-axis and a $y$-axis.
    • We will use the Cartesian plane, in which the $x$-axis is a horizontal line and the $y$-axis is a vertical line.
    • For the three values for $x$, let's choose a negative number, zero, and a positive number so we include points on both sides of the $y$-axis:
    • $\begin{aligned} (0)^{2}+y^{2} &= 100 \\ y^{2} &= 100 \\ \sqrt{y^{2}}&=\sqrt{100} \\ y &= \pm10 \end{aligned}$
    • $\begin{aligned} (6)^2+y^2&=100 \\ 36+y^2&=100 \\ 36+y^2-36&=100-36 \\ y^2&=64 \\ y&=\pm 8 \end{aligned}$

  • 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 $D$.
    • $y$-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 $D$.
    • $\displaystyle{\iint_D f(x,y)\ dx\, dy = \int_a^b dy \int_{\alpha (y)}^{ \beta (y)} f(x,y)\, dx}$
    • $D = \{ (x,y) \in \mathbf{R}^2 \ : \ x \ge 0, y \le 1, y \ge x^2 \}$

  • 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.  
    • To find the zero of a linear function algebraically, set $y=0$ and solve for $x$.
    • The blue line, $y=\frac{1}{2}x+2$, has a zero at $(-4,0)$; the red line, $y=-x+5$, has a zero at $(5,0)$.  

  • Graphing Quadratic Equations In Standard Form

    • The axis of symmetry for a parabola is given by:
    • Because $a=2$ and $b=-4,$ the axis of symmetry is:
    • More specifically, it is the point where the parabola intercepts the y-axis.
    • Note that the parabola above has $c=4$ and it intercepts the $y$-axis at the point $(0,4).$
    • The axis of symmetry is a vertical line parallel to the y-axis at  $x=1$.

  • 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).

  • Introduction to Ellipses

    • To do this, we introduce a scaling factor into one or both of the $x$-$y$ coordinates.
    • This has the effect of stretching the ellipse further out on the $x$-axis, because larger values of $x$ are now the solutions.
    • Now all the $y$ values are stretched vertically, further away from the origin.
    • The ellipse $x^2 +\left( \frac{y}{3} \right)^2 = 1$ has been stretched along the $y$-axis by a factor of 3 as compared to the circle $x^2 + y^2 = 1$.
    • The ellipse $\left( \frac{x}{3} \right)^2 +y^2 = 1$ has been stretched along the $x$-axis by a factor of 3 as compared to the circle $x^2 + y^2 = 1$.

  • Cylindrical Shells

    • In the shell method, a function is rotated around an axis and modeled by an infinite number of cylindrical shells, all infinitely thin.
    • 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:
    • 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:
    • $\displaystyle{V = 2\pi \int_a^b x \left | f(y) - g(y) \right | \,dy}$
    • Each segment located at $x$, between $f(x)$and the $x$-axis, gives a cylindrical shell after revolution around the vertical axis.