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:
    • $\begin{aligned} (0)^{2}+y^{2} &= 100 \\ y^{2} &= 100 \\ \sqrt{y^{2}}&=\sqrt{100} \\ y &= \pm10 \end{aligned}$
    • $\begin{aligned} (6)^2+y^2&=100 \\ 36+y^2&=100 \\ 36+y^2-36&=100-36 \\ y^2&=64 \\ y&=\pm 8 \end{aligned}$

  • 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  $x=1$.

  • 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 $x^2 + y^2 = 1$.

  • Symmetry of Functions

    • The image below shows an example of a function and its symmetry over the $x$-axis (vertical reflection) and over the $y$-axis (horizontal reflection).  
    • The axis splits the U-shaped curve into two parts of the curve which are reflected over the axis of symmetry.  
    • The function $y=x^2+4x+3$ shows an axis of symmetry about the line $x=-2$.  
    • Notice that the $x$-intercepts are reflected points over the axis of symmetry and are equidistant from the axis.
    • This type of symmetry is a translation over an axis.

  • Parts of an Ellipse

    • We will use the horizontal case to demonstrate how to determine the properties of an ellipse from its equation, so that $a$ is associated with x-coordinates, and $b$ with y-coordinates.
    • For a horizontal ellipse, that axis is parallel to the $x$-axis.
    • The major axis has length $2a$.
    • For a horizontal ellipse, it is parallel to the $y$-axis.
    • The minor axis has length $2b$.

  • 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.
    • A hyperbola aligned in this way is called an "East-West opening hyperbola. " Likewise, a hyperbola with its transverse axis aligned with the y-axis is called a "North-South opening hyperbola" and has equation:
    • The two thick black lines parallel to the conjugate axis (thus, perpendicular to the transverse axis) are the two directrices, D1 and D2.