vertices

Examples of vertices in the following topics:

• 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.

• Adding and Subtracting Vectors Using Components

  • Previously, we saw that vectors can be expressed in terms of their horizontal and vertical components .
  • This can be seen by adding the horizontal components of the two vectors ($4+4$) and the two vertical components ($3+3$).
  • These additions give a new vector with a horizontal component of 8 ($4+4$) and a vertical component of 6 ($3+3$).
  • To find the resultant vector, simply place the tail of the vertical component at the head (arrow side) of the horizontal component and then draw a line from the origin to the head of the vertical component.
  • It can be decomposed into a horizontal part and a vertical part as shown.

• Constant Acceleration

  • Analyzing two-dimensional projectile motion is done by breaking it into two motions: along the horizontal and vertical axes.
  • Because the acceleration due to gravity is along the vertical direction only, $a_x = 0$.
  • The velocity in the vertical direction begins to decrease as an object rises; at its highest point, the vertical velocity is zero.
  • As an object falls towards the Earth again, the vertical velocity increases again in magnitude but points in the opposite direction to the initial vertical velocity.
  • Throwing a rock or kicking a ball generally produces a projectile pattern of motion that has both a vertical and a horizontal component.

• 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:

• Components of a Vector

  • For two-dimensional vectors, these components are horizontal and vertical.
  • To find the vertical component, draw a line straight up from the end of the horizontal vector until you reach the tip of the original vector.
  • Decomposing a vector into horizontal and vertical components is a very useful technique in understanding physics problems.
  • Whenever you see motion at an angle, you should think of it as moving horizontally and vertically at the same time.
  • The vertical component stretches from the x-axis to the most vertical point on the vector.

• Reading Points on a Graph

  • The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges .Typically, a graph is depicted in diagrammatic form as a set of dots for the vertices, joined by lines or curves for the edges.
  • For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this is an indirected graph, because if person A shook hands with person B, then person B also shook hands with person A.
  • In contrast, if the vertices represent people at a party, and there is an edge from person A to person B when person A knows of person B, then this graph is directed, because knowledge of someone is not necessarily a symmetric relation (that is, one person knowing another person does not necessarily imply the reverse; for example, many fans may know of a celebrity, but the celebrity is unlikely to know of all their fans).
  • This latter type of graph is called a directed graph and the edges are called directed edges or arcs.Vertices are also called nodes or points, and edges are also called lines or arcs.

• Horizontal Asymptotes and Limits at Infinity

  • The asymptotes are computed using limits and are classified into horizontal, vertical and oblique depending on the orientation.
  • They can be computed using limits and are classified into horizontal, vertical and oblique asymptotes depending on the orientation.
  • Vertical asymptotes are vertical lines (perpendicular to the $x$-axis) near which the function grows without bound.
  • A common example of a vertical asymptote is the case of a rational function at a point $x$ such that the denominator is zero and the numerator is non-zero.

• Areas to the Left and Right of x

  • Remember, P ( X < x ) = Area to the left of the vertical line through x.
  • Area to the right of the vertical line through x