horizontal translation

Examples of horizontal translation in the following topics:

• Translations

  • A translation of a function is a shift in one or more directions.
  • In algebra, this essentially manifests as a vertical or horizontal shift of a function.
  • To translate a function horizontally is the shift the function left or right.
  • The general equation for a horizontal shift is given by:
  • Let's use the same basic quadratic function to look at horizontal translations.

• 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).  
  • A function can have symmetry by reflecting its graph horizontally or vertically.  
  • This type of symmetry is a translation over an axis.

• The Existence of Inverse Functions and the Horizontal Line Test

  • Recognize whether a function has an inverse by using the horizontal line test

• Asymptotes

  • There are three kinds of asymptotes: horizontal, vertical and oblique.
  • Horizontal asymptotes of curves are horizontal lines that the graph of the function approaches as $x$ tends to $+ \infty$ or $- \infty$.
  • Horizontal asymptotes are parallel to the $x$-axis.
  • The $x$-axis is a horizontal asymptote of the curve.
  • Hence, horizontal asymptote is given by:

• One-to-One Functions

  • If any horizontal line intersects the graph at more than one point, the function is not one-to-one.
  • One way to check if the function is one-to-one is to graph the function and perform the horizontal line test.  
  • The graph below shows that it forms a parabola and fails the horizontal line test.
  • Notice it fails the horizontal line test.
  • Because the horizontal line crosses the graph of the function more than once, it fails the horizontal line test and cannot be one-to-one.

• Stretching and Shrinking

  • Now lets analyze horizontal scaling. 
  • This leads to a "shrunken" appearance in the horizontal direction.
  • In general, the equation for horizontal scaling is:
  • If $c$ is greater than one the function will undergo horizontal shrinking, and if $c$ is less than one the function will undergo horizontal stretching.
  • If we want to induce horizontal shrinking, the new function becomes:

• 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.
  • For a horizontal ellipse, it is parallel to the $y$-axis.
  • For a horizontal ellipse, the foci have coordinates $(h \pm c,k)$, where the focal length $c$ is given by
  • This diagram of a horizontal ellipse shows the ellipse itself in red, the center $C$ at the origin, the focal points at $\left(+f,0\right)$ and $\left(-f,0\right)$, the major axis vertices at $\left(+a,0\right)$ and $\left(-a,0\right)$, the minor axis vertices at $\left(0,+b\right)$ and $\left(0,-b\right)$.

• Restricting Domains to Find Inverses

  • Without any domain restriction, $f(x)=x^2$ does not have an inverse function as it fails the horizontal line test.
  • But if we restrict the domain to be $x > 0$ then we find that it passes the horizontal line test and therefore has an inverse function.  
  • Notice that the parabola does not have a "true" inverse because the original function fails the horizontal line test and must have a restricted domain to have an inverse.
  • This function fails the horizontal line test, and therefore does not have an inverse.
  • However, if we restrict the domain to be $x>0$, then we find that it passes the horizontal line test and will match the inverse function.

• Transformations of Functions

  • The four main types of transformations are translations, reflections, rotations, and scaling.
  • A translation moves every point by a fixed distance in the same direction.
  • One possible translation of $f(x)$ would be $x^3 + 2$.  
  • This would then be read as, "the translation of $f(x)$ by two in the positive y direction".
  • The function $f(x)=x^3$ is translated by two in the positive $y$ direction (up).

• Reflections

  • A horizontal reflection is a reflection across the $y$-axis, given by the equation: