Algebra I

Determining the Equation of a Line

Big Picture

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.

Key Terms

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.

Slope-Intercept Form

The slope-intercept form: \(y = mx+b\)

Given the Slope and y-Intercept

  • Plug in the slope of the line for m and the y-coordinate of the y-intercept for \(b\)

Given the Slope and a Point

  • Plug in the slope of the line for m
  • Plug in the coordinates of the point into \(x\) and y and solve for \(b\)
  • Rewrite the equation with the values for m and b plugged in

Given Two Points

  • Use the two points to determine the slope: \(x = \frac{y_2 - y_1}{x_2 - x_1}\)
  • Plug in the slope for \(m\)
  • Plug in the coordinates of one of the points into \(x\) and \(y\) and solve for \(b\)
  • Rewrite the equation with the values for \(m\) and \(b\) plugged in

Point-Slope Form

The point-slope form: \(y-y_{0} = m(x - x_{0})\)

Given the Slope and y-Intercept

  • Plug in the slope of the line for \(m\) and the coordinates of the \(y-intercept\) for \(x_0\) and \(y\)
  • \(x_0\) is equal to \(0\), so the point-slope form of the equation simplifies to \(y-y_0 = mx\)

Given the Slope and a Point

  • Plug in the slope of the line for and the coordinates of the point for \(x_0\) and \(y_0\)

Given Two Points

  • Use the two points to determine the slope: \(x = \frac{y_2 - y_1}{x_2 - x_1}\)
  • Plug in the slope for and the coordinates for one of the points for \(x_0\) and \(y_0\)

Standard Form

  • The standard form: \(Ax + By = C\)
  • To write an equation in standard form, first write the equation in slope-intercept form
  • Rearrange so that the variables are on one side of the equation and the constant on the other.
  • If any of the coefficients are fractions, multiply by the least common denominator so that all the coefficients are whole numbers.

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Algebra I

Determining the Equation of a Line cont.

Parallel & Perpendicular Lines

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

  • If m is the slope of one line, the perpendicular line has the slope of \(- \frac{1}{m}\)
  • \(m . - \frac{1}{m} = -1\)

Equations of Parallel and Perpendicular Lines

To find the equation for a line \(M\) parallel to a given line \(L\):

Slope-Intercept Form

  • Find the slope of \(L\) (the given line). \(M\) will have the same slope as \(L\) does.
  • Put the slope into the slope-intercept form: \(y = mx + b\).
  • If line \(L\) must go through a specific point, plug it into the slope-intercept form to find \(b\text{.}\)
  • If not, any value of \(b\) will work.

To find the equation for a line \(K\) perpendicular to a given line \(L\):

  • Find the slope of \(L\) (the given line).
  • . Find the slope of the perpendicular line (the negativereciprocal of the slope of line L). Plug the slope into the slopeintercept form.
  • If line \(K\) must go through a specific point, plug it into the slope-intercept form to find \(b\).
  • If not, any value of \(b\) will work.

Perpendicular lines

Linear Model

Fitting a Line to Data Points

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,

  • Plot the data points on a graph.
  • Draw the best fit line through the data points by eyeballing it.
  • Find the slope using two points.
  • Using the slope and a data point, write the equation.

Predicting with Linear Models

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.

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