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 xx or $y$-axis.
    • The reflection of a function can be performed along the xx-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 xx-axis, $y$-axis, or the line $y=x$

  • The Cartesian System

    • The horizontal axis is known as the xx-axis, and the vertical axis is known as the $y$-axis.
    • Each point can be represented by an ordered pair $(x,y) ,wherethe, where the x−coordinateisthepoint′sdistancefromthe-coordinate is the point's distance from the y$-axis and the $y−coordinateisthedistancefromthe-coordinate is the distance from the x$-axis.
    • On the xx-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)(-1.5,-2.5) lie between -1 and -2 on the xx-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 xx-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)(0,1) and is increasing as xx takes on larger and larger values.
    • As you can see in the graph below, the graph of $y=\frac{1}{2}^xissymmetrictothatof is symmetric to that of y=2^xoverthe over the y$-axis.
    • That is, if the plane were folded over the $y$-axis, the two curves would lie on each other.
    • The function $y=b^xhasthe has the x$-axis as a horizontal asymptote because the curve will always approach the xx-axis as xx 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)(0,1) and increases as xx approaches infinity.

  • Graphing Equations

    • For an equation with two variables, xx and $y,weneedagraphwithtwoaxes:an, we need a graph with two axes: an x$-axis and a $y$-axis.
    • We will use the Cartesian plane, in which the xx-axis is a horizontal line and the $y$-axis is a vertical line.
    • For the three values for xx, 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 DD onto either the xx-axis or the $y$-axis is bounded by the two values, aa and bb.
    • xx-axis: If the domain DD is normal with respect to the xx-axis, and f:D→Rf:D \to R is a continuous function, then α(x)\alpha(x)  and β(x)\beta(x) (defined on the interval [a,b][a, b]) are the two functions that determine DD.
    • $y$-axis: If DD is normal with respect to the $y$-axis and f:D→Rf:D \to R is a continuous function, then $\alpha(y)$ and $\beta(y)$ (defined on the interval $[a, b])arethetwofunctionsthatdetermine) are the two functions that determine D$.
    • $\displaystyle{\iint_D f(x,y)\ dx\, dy = \int_a^b dy \int_{\alpha (y)}^{ \beta (y)} f(x,y)\, dx}$
    • $D = \{ (x,y) \in \mathbf{R}^2 \ : \ x \ge 0, y \le 1, y \ge x^2 \}$

  • Zeroes of Linear Functions

    • Graphically, where the line crosses the xx-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 xx-axis.  
    • If the horizontal line overlaps the xx-axis, (goes through the $y$-axis at zero) then there are infinitely many zeros, since the line intersects the xx-axis multiple times.  
    • To find the zero of a linear function algebraically, set $y=0$ and solve for $x$.
    • The blue line, $y=\frac{1}{2}x+2,hasazeroat, has a zero at (-4,0);theredline,; the red line, y=-x+5,hasazeroat, has a zero at (5,0)$.  

  • Graphing Quadratic Equations In Standard Form

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

  • Area of a Surface of Revolution

    • If the curve is described by the function $y = f(x) (a≤x≤b),thearea, the area A_yisgivenbytheintegral is given by the integral A_x = 2\pi\int_a^bf(x)\sqrt{1+\left(f'(x)\right)^2} \, dxforrevolutionaroundthe for revolution around the x$-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)x(t), $y(t),with, with t$ 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 xx-axis, and provided that $y(t)$ is never negative, the area is given by:
    • A portion of the curve x=2+coszx=2+\cos z rotated around the zz-axis (vertical in the figure).

  • Introduction to Ellipses

    • To do this, we introduce a scaling factor into one or both of the xx-$y$ coordinates.
    • This has the effect of stretching the ellipse further out on the xx-axis, because larger values of xx 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)f(x) and g(x)g(x) and the lines x=ax=a and x=bx=b about the $y$-axis is given by:
    • The volume of solid formed by rotating the area between the curves of $f(y)andandthelines and and the lines y=aand and y=baboutthe about the x$-axis is given by:
    • $\displaystyle{V = 2\pi \int_a^b x \left | f(y) - g(y) \right | \,dy}$
    • Each segment located at xx, between f(x)f(x)and the xx-axis, gives a cylindrical shell after revolution around the vertical axis.