Examples of boundary line in the following topics:
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- 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:
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- 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.
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- Recognize whether a function has an inverse by using the horizontal line test
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- 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
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- 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.
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- 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 $\frac{4}{5}$.
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- 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).
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- 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.
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- 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 $5$.