vertical translation

Examples of vertical 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 vertically is to shift the function up or down.
  • In general, a vertical translation is given by the equation:
  • Let's use a basic quadratic function to explore vertical 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 Vertical Line Test

  • The vertical line test is used to determine whether a curve on an $xy$-plane is a function
  • If, alternatively, a vertical line intersects the graph no more than once, no matter where the vertical line is placed, then the graph is the graph of a function.
  • The vertical line test demonstrates that a circle is not a function.
  • Thus, it fails the vertical line test and does not represent a function.
  • Any vertical line in the bottom graph passes through only once and hence passes the vertical line test, and thus represents a function.

• Parts of a Hyperbola

  • The vertices have coordinates $(h + a,k)$ and $(h-a,k)$.
  • The line connecting the vertices is called the transverse axis.
  • The major and minor axes $a$ and $b$, as the vertices and co-vertices, describe a rectangle that shares the same center as the hyperbola, and has dimensions $2a \times 2b$.
  • The rectangle itself is also useful for drawing the hyperbola graph by hand, as it contains the vertices.
  • The vertices have coordinates $(h+\sqrt{2m},k+\sqrt{2m})$ and $(h-\sqrt{2m},k-\sqrt{2m})$.

• Asymptotes

  • There are three kinds of asymptotes: horizontal, vertical and oblique.
  • Vertical asymptotes are vertical lines near which the function grows without bound.
  • The $y$-axis is a vertical asymptote of the curve.
  • Vertical asymptotes occur only when the denominator is zero.
  • Therefore, a vertical asymptote exists at $x=1$.

• Stretching and Shrinking

  • First, let's talk about vertical scaling.  
  • This leads to a "stretched" appearance in the vertical direction.
  • In general, the equation for vertical scaling is:
  • If $b$ is greater than one the function will undergo vertical stretching, and if $b$ is less than one the function will undergo vertical shrinking.
  • If we want to vertically stretch the function by a factor of three, then the new function becomes:

• 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).

• Parts of an Ellipse

  • if the ellipse is oriented vertically.
  • For a vertical ellipse, the association is reversed.
  • Its endpoints are the major axis vertices, with coordinates $(h \pm a, k)$.
  • Its endpoints are the minor axis vertices, with coordinates $(h, k \pm b)$.
  • 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 $\left(0,-b\right)$.

• Reflections

  • A vertical reflection is a reflection across the $x$-axis, given by the equation:

• What is a Linear Function?

  • Vertical lines have an undefined slope, and cannot be represented in the form $y=mx+b$, but instead as an equation of the form $x=c$ for a constant $c$, because the vertical line intersects a value on the $x$-axis, $c$.  
  • Vertical lines are NOT functions, however, since each input is related to more than one output.