The equation of the line can reveal a lot about the function it represents. The slope of the equation represents how fastand in what direction the function is moving. The intercept of the equation reveals how the function relates to the axes.The equation of a particular line can also help predict future values for that function.
Slope-Intercept Form: A form of writing a linear equation in two variables: \(y = mx + b\), where m is the slope, \(b\) is they-intercept, and \(x\) and \(y\) are the variables.
Slope: A ratio of the distance moved vertically over the distance moved horizontally in a non-vertical line.
y-Intercept: The point where a line crosses the y-axis
Parallel Lines: Lines that do not intersect or touch at a point. Parallel lines have the same slope.
Perpendicular Lines: Lines that intersect or touch at one point. Perpendicular lines have slopes that are negative reciprocals of each other.
The slope-intercept form: \(y = mx+b\)
The point-slope form: \(y-y_{0} = m(x - x_{0})\)
Parallel lines must have the same slope.
Any two lines with identical slopes are parallel.
The diagram on the right shows five parallel lines. Changing the y-intercept (the \(b\) in the slope-intercept form) only moves(translates) the graph up and down. The lines never intersect.
Perpendicular lines intersect at one point and must have slopes that are negative reciprocals of each other
To find the equation for a line \(M\) parallel to a given line \(L\):
To find the equation for a line \(K\) perpendicular to a given line \(L\):
When given a table of data points that are ordered pairs in the form of \((x, y)\), the data points can be graphed on a scatter plot.
Oftentimes the data points won’t fall perfectly on a straight line. Instead, these points may look like they are “scattered” around a straight line.
To find this line,
By finding a linear equation for a data set, you can predict future data points by plugging in the \(x\) or \(y\) values that you want into the equation.
The red line in the graph is the best fit line for the data points.