## P.2 Linear Models and Rates of Change

### 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 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

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

### 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. We’re just treating it that way to come up with an average.

### 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 $−2/3$.