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.
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 ignored a lot in earlier math, but it’s the easiest to use because all you need is the slope and a single point. And typically, if you have the slope, you have two points to choose from. You can then take point-slope form and rearrange it as necessary.
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
\[\begin{align*} y+2&=3x-3 \\ y&=3x−5 \end{align*}\]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. We’re just treating it that way to come up with an average.
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 $−2/3$.