• Find the slope of a line passing through two points.
  • Write the equation of a line with a given point and slope.
  • Interpret slope as a ration or as a rate in a real-life application.
  • Sketch the graph of a linear equation in slope-intercept form.
  • Write equations of lines that are parallel or perpendicular to a given line.

Assignment

  • Vocabulary and teal boxes
  • p20 10, 11, 22, 23, 28, 31, 33, 37, 43, 47, 57, 59, 64, 67, 73–75, 77

The Slope of a Line

Rise over run. Average rate of change. Change in $y$ over change in $x$. $Δy/Δx$. $(f(a)−f(b))/(a−b)$.

Slope is a popular topic in calculus, despite being introduced back in Algebra I. Make sure you know how to find it, and that you need two points in order to do so.

Equations of Lines

You’ve been shown a lot of different forms lines, but we only care about one in this class. Point-slope form.

\[y−y_1=m(x−x_1)\]

It gets often ignored in earlier math, but because all you need is the slope and a single point, it’s the easiest to use. You can rearrange it as necessary after that.

Quick example, say you are given some line that has a slope of 3 and a point of $(1,−2)$. What would the equation be?

\[y−(−2)=3(x−1)\]

And if you wanted slope-intercept form

\[\begin{align*} y+2&=3x-3 \\ y&=3x−5 \end{align*}\]

Ratios and Rates of Change

Since slope is a ratio, it’s handy for real-world applications as well. The population of Utah was 2,427,000 in 2005 and 3,161,000 in 2018. Treating time as $x$, which is customary, our change in population over change in time would give us the average rate of change, which would be a growth of 56,462 people per year.

Keep in mind that the actual population growth was likely not linear. This quick and dirty method gives us an average rate of change. Finding a more precise measurement is part of what you’ll learn this year.

Parallel and Perpendicular Lines

Parallel lines have an equal slope. Perpendicular lines have opposite reciprocal slopes. If one line has a slope of $2/3$, a line perpendicular would have a slope of $−2/3$.