vertical line test

Examples of vertical line test 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
  • Apply the vertical line test to determine which graphs represent functions.
  • 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.

• Introduction to Domain and Range

  • We can also tell this mapping, and set of ordered pairs is a function based on the graph of the ordered pairs because the points do not make a vertical line.  
  • If an $x$value were to repeat there would be two points making a graph of a vertical line, which would NOT be a function.  
  • This mapping or set of ordered pairs is a function because the points do not make a vertical line.  
  • This is called the vertical line test of a function.  

• Introduction to Inverse Functions

  • This is equivalent to reflecting the graph across the line $y=x$, an increasing diagonal line through the origin.
  • Even though the blue (function) curve is a function (passes the vertical line test), its inverse (red) only includes the positive square root values and not the negative square root values of the functions range.  
  • Test to make sure this solution fills the definition of an inverse function.
  • The function graph (red) and its inverse function graph (blue) are reflected about the line $y=x$ (dotted black line)  Notice that any ordered pair on the red curve has its reversed ordered pair on the blue line.  
  • The black line represents the line of reflection, in which is $y=x$.

• The Existence of Inverse Functions and the Horizontal Line Test

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

• 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 asymptotes of the hyperbola are straight lines that are the diagonals of this rectangle.
  • We can therefore use the corners of the rectangle to define the equation of these lines:
  • Then draw in the asymptotes as extended lines that are also the diagonals of the rectangle.

• Asymptotes

  • In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity.
  • Vertical asymptotes are vertical lines near which the function grows without bound.
  • These are diagonal lines so that the difference between the curve and the line approaches 0 as $x$ tends to $+ \infty$ or $- \infty$.
  • Therefore, a vertical asymptote exists at $x=1$.
  • The graph of a function with a horizontal ($y=0$), vertical ($x=0$), and oblique asymptote (blue line).

• What is a Linear Function?

  • Also, its graph is a straight line.
  • 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.
  • The blue line, $y=\frac{1}{2}x-3$ and the red line, $y=-x+5$ are both linear functions.  
  • The blue line has a positive slope of $\frac{1}{2}$ and a $y$-intercept of $-3$; the red line has a negative slope of $-1$ and a $y$-intercept of $5$.

• Slope

  • Slope describes the direction and steepness of a line, and can be calculated given two points on the line.
  • The direction of a line is either increasing, decreasing, horizontal or vertical.
  • If a line is vertical the slope is undefined.
  • Slope is calculated by finding the ratio of the "vertical change" to the "horizontal change" between any two distinct points on a line.
  • Count the rise on the vertical leg of the triangle: 4 units.

• Reflections

  • For this section we will focus on the two axes and the line $y=x$.
  • A vertical reflection is a reflection across the $x$-axis, given by the equation:
  • The third type of reflection is a reflection across a line.  
  • Let's look at the case involving the line $y=x$.  
  • The reflected equation, as reflected across the line $y=x$, would then be:

• One-to-One Functions

  • An easy way to check if a function is a one-to-one is by graphing it and then performing the horizontal line test.
  • 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.