Number of vocabulary words: **4**

This section is a refresher on a some concepts from Algebra I. It’s dense, but hopefully a lot of this is old knowledge that just needs some dusting off.

The set of all possible inputs is its domain, while the valid outputs are the range. Typically, these are $x$ and $y$, respectively, but can be labeled otherwise depending on the scenario.

For most of the functions you’ve seen so far, the domain and range have no restrictions and include all real numbers. The exceptions are your square root function and the rational function, but we’ll get into those later in the course.

But, when dealing with real-world applications, your domain and range are now restricted by reality. So, a function relating distance traveled and fuel consumed would not include any negative numbers in either the domain or range.

When talking about intervals of numbers, you’ll see either set or interval notation used to save on the number of words someone has to type.

Set-builder notationuses a verbal description or an inequality to describe the numbers.

Interval notationrepresents a set of real numbers by its minimum and maximum boundaries.

For example, to describe all non-negative numbers in set-builder, it would look like

\[\{x | x \ge 0\}\]Interval notation omits the variable, and uses brackets to indicate lesser/greater or equal and parentheses for lesser/greater than. The infinity symbol is used when there is no bound on one side. So, here’s non-negative numbers again.

\[[0,\infty)\]Use whatever you are comfortable with, as long as it’s clear what you mean.

The intercepts of any function are where the other variable is zero. So, $(0,4)$ is a $y$-intercept and $(-3,0)$ is an $x$-intercept.

The $x$-intercepts are also the

zeros of the functionbecause they are the input values that result in a function output value of 0.

The first step to finding intercepts is to plug 0 in to the other variable. You can find the $y$-intercept of $y=6x^{3}-2x^{2}-x+2$ by plugging in 0 for every $x$. Finding the $x$-intercept requires that you take an algebra course.

Positive and negative intervals are where the function is positive or negative. If the $y$-values are above the $x$-axis, it’s positive. Below is negative.

This goes hand in hand with finding the zeros of a function, as those will indicate at what point a function crosses over the $x$-axis.

Increasing and decreasing intervals are also pretty self-explanatory, though worth pointing out is that the direction changes at local maximums and minimums.

You will see functions with several local extrema, with one of them sometimes being the absolute maximum or minimum.

Finding those extrema is a whole topic on its own … in Calculus. For now, graphing or creating a table will get the job done.

Rate of change is just how one variable changes compared to another, like miles per hour.

The

average rate of changeis the ratio $\frac{f(a)-f(b)}{a-b}$.

Hopefully that looks familiar, because it’s just slope, which is sometimes written as $\Delta y / \Delta x$. So, regardless of what the function looks like, if you want the average rate of change of an interval, pretend there’s a straight line connecting the two points at either end.

What’s the average rate of change of $f(x)=x^2$ over the interval $[-2,3]$? First, get the points at the ends of the interval. This will get you $(-2,4)$ and $(3,9)$. Now we just pretend there is a straight line connecting these two, and find the slope.

\[\frac{9-4}{3+2} = \frac{5}{5} = 1\]