x-axis

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.

• 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.

• A Graphical Interpretation of Quadratic Solutions

  • Notice that the parabola intersects the $x$-axis at two points: $(-1, 0)$ and $(2, 0)$.
  • Looking at the graph of the function, we notice that it does not intersect the $x$-axis.
  • Graph of the quadratic function $f(x) = x^2 - x - 2$
  • Graph showing the parabola on the Cartesian plane, including the points where it crosses the x-axis.
  • Recognize that the solutions to a quadratic equation represent where the graph of the equation crosses the x-axis

• Introduction to Ellipses

  • Let's start by dividing all $x$ coordinates by a factor $a$, and therefore scaling the $x$ values.
  • We simply substitute $\displaystyle{\frac{x}{a}}$ into the equation instead of $x$.
  • This has the effect of stretching the ellipse further out on the $x$-axis, because larger values of $x$ are now the solutions.
  • 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 $x^2 + y^2 = 1$.

• Asymptotes

  • Horizontal asymptotes are parallel to the $x$-axis.
  • They are parallel to the $y$-axis.
  • However, no matter how large $x$ becomes, $\frac {1}{x}$ is never $0$, so the curve never actually touches the $x$-axis.
  • The $x$-axis is a horizontal asymptote of the curve.
  • The $y$-axis is a vertical asymptote of the curve.

• 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.

• 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.

• Solving Problems with Rational Functions

  • We can use algebraic methods to calculate their $x$-intercepts (also known as zeros or roots), which are points where the graph intersects the $x$-axis.
  • For $f(x) = \frac{P(x)}{Q(x)}$, if $P(x) = 0$, then $f(x) = 0$.
  • $\begin {aligned} 0 &=x^2 - 3x + 2 \\&= (x - 1)(x - 2) \end {aligned}$
  • $\begin {aligned} 0&=x^3 - 2x \\&= x(x^2 - 2) \end {aligned}$
  • $x^2 - 2 = 0 \\ x^2 = 2 \\ x = \pm \sqrt{2}$