boundary line

Examples of boundary line in the following topics:

• Graphing Inequalities

  • The straight line shown is called a boundary line.
  • This is called the boundary line.
  • If the inequality is $<$ or $>$, draw the boundary line dotted.
  • First, we need to graph the boundary line.
  • This gives the boundary line below:

• Nonlinear Systems of Inequalities

  • Systems of nonlinear inequalities can be solved by graphing boundary lines.
  • Every inequality has a boundary line, which is the equation produced by changing the inequality relation to an equals sign.
  • The boundary line is drawn as a dashed line (if $<$ or $>$ is used) or a solid line (if $\leq$ or $\geq$ is used).
  • One side of the boundary will have points that satisfy the inequality, and the other side will have points that falsify it.
  • Graphing both inequalities reveals one region of overlap: the area where the parabola dips below the line.

• The Distance Between Two Lines

• The Existence of Inverse Functions and the Horizontal Line Test

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

• Parallel and Perpendicular Lines

  • Two lines in a plane that do not intersect or touch at a point are called parallel lines.
  • This means that if the slope of one line is $m$, then the slope of its perpendicular line is $\frac{-1}{m}$.
  • The value of the slope will be equal to the current line, since the new line is parallel to it.  
  • The line $f(x)=3x-2$ in red is perpendicular to line $g(x)=\frac{-1}{3}x+1$ in blue. 
  • Write equations for lines that are parallel and lines that are perpendicular

• The Vertical Line Test

  • To use the vertical line test, take a ruler or other straight edge and draw a line parallel to the $y$-axis for any chosen value of $x$.
  • 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.
  • For example, a curve which is any straight line other than a vertical line will be the graph of a function.
  • The vertical line test demonstrates that a circle is not 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.

• Slope

  • Slope describes the direction and steepness of a line, and can be calculated given two points on the line.
  • In mathematics, the slope of a line is a number that describes both the direction and the steepness of the line.
  • Putting the equation of a line into this form gives you the slope ($m$) of a line, and its $y$-intercept ($b$).
  • In other words, a line with a slope of $-9$ is steeper than a line with a slope of $7$.
  • The slope of the line is $\frac{4}{5}$.

• Graphs of Linear Inequalities

  • If it is $>$ or $<$, then use a dotted or dashed line, since ordered pairs found on the line would result in a false statement.
  • Since the equation is less than or equal to, start off by drawing the line $y=x+2$, using a solid line.  
  • All possible solutions are shaded, including the ordered pairs on the line, since the inequality is $\leq$ the line is solid.  
  • There are no solutions above the line.
  • The overlapping shaded area is the final solution to the system of linear inequalities because it is comprised of all possible solutions to $y<-\frac{1}{2}x+1$ (the dotted red line and red area below the line) and $y\geq x-2$ (the solid green line and the green area above the line).  

• Zeroes of Linear Functions

  • The graph of a linear function is a straight line.
  • If there is a horizontal line through any point on the $y$-axis, other than at zero, there are no zeros, since the line will never cross the $x$-axis.  
  • All lines, with a value for the slope, will have one zero.  
  • The blue line, $y=\frac{1}{2}x+2$, has a zero at $(-4,0)$; the red line, $y=-x+5$, has a zero at $(5,0)$.  
  • Since each line has a value for the slope, each line has exactly one zero.

• 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 $5$.