vertical line test

(noun)

A visual test that determines whether a curve is a function or not by examining the number of $y$-values associated with each $x$-value that lies on the curve.

Related Terms

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.  

  • Homogeneity and Heterogeneity

    • Imagine that you have a scatter plot, on top of which you draw a narrow vertical strip.
    • Another way of putting this is that the prediction errors will be similar along the regression line.
    • When a scatter plot is heteroscedastic, the prediction errors differ as we go along the regression line.
    • Similarly, in testing for differences between sub-populations using a location test, some standard tests assume that variances within groups are equal.
    • Drawing vertical strips on top of a scatter plot will result in the $y$-values included in this strip forming a new data set.

  • Least-Squares Regression

    • Any other potential line would have a higher SSE than the best fit line.
    • Therefore, this best fit line is called the least squares regression line.
    • Here is a scatter plot that shows a correlation between ordinary test scores and final exam test scores for a statistics class:
    • This method minimizes the sum of squared vertical distances between the observed responses in the dataset and the responses predicted by the linear approximation.
    • This graph shows the various scattered data points of test scores.

  • The Existence of Inverse Functions and the Horizontal Line Test

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

  • Making a Box Model

    • When we explore relationships between multiple variables, even more statistics arise, such as the coefficient estimates in a regression model or the Cochran-Maentel-Haenszel test statistic in partial contingency tables.
    • A multitude of statistics are available to summarize and test data.
    • The box lies on a vertical axis in the range of the sample.
    • A common version is to place a horizontal line at the median, dividing the box into two.
    • Another common extension of the box model is the 'box-and-whisker' plot , which adds vertical lines extending from the top and bottom of the plot to, for example, the maximum and minimum values.

  • Adding and Subtracting Vectors Using Components

    • Previously, we saw that vectors can be expressed in terms of their horizontal and vertical components .
    • 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.
    • This new line is the resultant vector.
    • It can be decomposed into a horizontal part and a vertical part as shown.

  • 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).

  • 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.
    • Horizontal asymptotes are horizontal lines that the graph of the function approaches as $x$ tends toward $+ \infty$ or $- \infty$.
    • Vertical asymptotes are vertical lines (perpendicular to the $x$-axis) near which the function grows without bound.
    • Oblique asymptotes are diagonal lines so that the difference between the curve and the line approaches $0$ as $x$ tends toward $+ \infty$ or $- \infty$.