Graph of y=x^5+4x^4-2x^2+1

Welcome to Precalculus. This page is a guide to the Precalculus topics that are blogged about on this site.

Functions, generally

Precalculus is mostly about functions, so it makes sense to explore what a function is. A function is a relation between two things, call them x and y, for which every x has one and only one corresponding y. For example, in a relation if x is the year and y is the high temperature for the year in a certain place, then this relation is a function, because each place should have just one high temperature for a year. On the other hand, if x is, again, the year and this time y is all the temperatures recorded in a certain place, then that relation is not a function, because there will be many temperatures recorded over the course of a year, so each value of x has more than one value of y.

On a graph, a function is a relation that passes the vertical line test. That means that any vertical line that crosses the graph will cross at one point only. That’s a quick way to identify a function. Here are some examples of applying the vertical line test.

A function’s domain is the set of values x can take and its range is the set of values y can take. This post shows an example of how to find a function’s range.

Polynomial functions

One class of functions that you may see a lot of in Precalculus is polynomials.

When you study polynomials, you are typically interested in where they cross the x-axis, when they change from increasing to decreasing and vice versa, and what happens at very large values of x or -x. That last issue is called end behavior.

Think back to when you studied quadratics. You might get an equation like x^2+2x=-1. What’s the first step in solving it? Get everything of substance onto the left-hand side. Then the right-hand side equals 0. After that, if you can, factor the left-hand side and set each factor equal to 0. You can also solve some higher-degree polynomial equations by factoring, but it gets more challenging as the equations get more complicated. So we find some rules to help us.

Where does the graph cross the x-axis?

Graph of y=x^5+4x^4-2x^2+1A x-value where a graph crosses the x-axis can be called an x-intercept or a root or a solution (because it represents the solution to an equation). Often your job will be to find those x-values.

How many x-intercepts do you need to look for? The maximum number zeros of a polynomial function is equal to the function’s degree. What are those x-values? The rational roots test can give you some clues.

End behavior


When x or -x gets very large, does a polynomial’s value approach negative infinity or positive infinity? That depends on the function’s degree and on the sign of its leading term.

Other issues

You may be given solutions and asked to find the function that has those solutions.

Rational functions

In Precalculus you’ll also study rational functions, which are fractions with variables in their denominators.

Graph of a slant asymptote

A rational expression may have vertical, horizontal, or slant (aka oblique) asymptotes. An asymptote is a line that a graph approaches but never reaches.

  • A zero denominator may cause a vertical asymptote.
  • If the numerator and denominator have equal degrees, there is a horizontal asymptote.
  • If the numerator’s degree is less than the denominator’s, there is a horizontal asymptote at y=0.
  • If the numerator’s degree is exactly one more than the denominator’s, there is a slant asymptote.
  • If the numerator’s degree is more than one more than the denominator’s, there is no asymptote.

A polynomial function of x divided by a binomial is an example of a rational function, for example \frac{x^2+2x+1}{x-1} . The remainder theorem says that when a polynomial function of x is divided by x-c, where c is any real number, the quotient equals the function’s value at point c.  With the remainder theorem you can use division — that is polynomial long division or synthetic divisionto find a function’s remainder at a point; you can also use it to find a function’s value at a point from its remainder there. For example, to find the value of the function \fn_jvn f(x)=x^4 +10x^3 +35x^2 +50x+24 at x=-5, divide \fn_jvn \small x^4+10x^3+35x^2+50x+24  by x+5. That will give you a remainder of 24. Thus f(-5) = 24.

Exponential Functions

Exponential growth is an increase in something — bacteria, money, population, whatever — that is in proportion to the amount present. Here is one example of exponential growth. Here is another.

But things don’t always increase; sometimes they decrease. If the decrease is in proportion to the amount present, that’s exponential decay. Here is an example of exponential decay

An exponential equation is an equation with variables in one or more exponents. Here are some thoughts about how to solve an exponential equation.

Logarithmic functions

A logarithm is kind of a funny animal; it doesn’t always do what you might imagine. This is illustrated in the graph of the logarithm of a product.

Sliding a function sideways

f(x+1) graphs up 1 unit to the left of f(x).

f(x+1) graphs up 1 unit to the left of f(x).

Let’s say you’re looking at the graph of a function f(x). Then what does f(x+1) look like? Imagine yourself standing at the origin of the coordinate plane, and say the function at x=o, where you are, is above your head and you are looking up at it. And you wonder what’s happening one unit in front of you, at x=1. You, at x=0, are looking at the y-value for x=1. Since you’re at x=0, you replace the y-value that’s already there (in green)  with the y-value you found for x=1 — that is, with f(1) (in blue).

What if you do that for every x-value on the graph: At every point, replace the y-value with the y-value you see for x+1. Then, at every point, you are replacing f(x) with the y-value that is one point to the right. Thus, f(x+1) is one unit to the left of f(x), no matter how much that plus sign makes you want to move it to the right. Similarly, f(x-1) is f(x) moved one unit to the right. See more about horizontal translations here. See more about horizontal translations here.