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Math SATs- Basic Functions

Welcome to a Math SAT Preparation Lesson. In this follow-up lesson to Analytic Geometry we will take a look at basic non-linear functions.

As we’ve already seen, we can use x’s and y’s to represent lines. It’s a given that if we join several lines, we can then create polygons like triangles and rectangles all on a mathematical grid, and use equations to represent them on the XY plane. You may ask yourself now, can we draw cooler shapes too?

Yeah, we can, otherwise there wouldn’t be much point to this lesson.

Let’s start with my personal favorite- the circle.

We all know what a circle looks like (though it's difficult to draw one). We can use equations to represent circles. There are 2 important parts to a circle- the center and the radius. So, a general equation for a circle with center (a,b) and radius r is: . The simplest circle centered at the origin with radius 1 has the equation . It's that simple. And just like last time, we can find intersections of circles with lines. Note though that if you draw a line through a circle, the line will hit the edge twice, which makes sense, since the degree of x in the circle equation is 2.

A shape that for some reason math teacher like to teach you in high school, even though it has little practical use at that stage, is a parabola.

Parabolas are hard to describe, but I’ll give it a shot. Look at this picture: . The smile looks sort of like a parabola. Parabolas can also be upside down, like a frowning smiley face. Also note that this particular picture also contains a circle, so it’s good. After you’re done this lesson, try to write equations to create a smiley face on a grid.

In the meanwhile, what you need to know about parabolas are a few things- the vertex, which is pretty much the center of the parabola from which it starts to take shape, and the so-called “stretch-factor”, which shows how fat the parabola is. A parabola with vertex at (a,b) with a stretch factor of k has a (vertex-form) equation of . An important thing to know is that when the stretch factor k is positive, the parabola is happy, or it opens upwards. When k is negative the parabola is sad and upside-down.

Now, let’s introduce the concept for which we all came for- functions.

A function is a relationship that equips values of x to appropriate values of y. The important thing about functions is that for each x there is only one value of y, so you can have (2,4) and (3,4) in a function, but not (2,6) and (2,5). So a circle isn’t a real function, since for each y value we have 2 possible x values. However, lines and parabolas, as well as cubic and higher powered functions are real functions.

How do functions work?

Let’s say we’ve got the following set of values:

x	-1	0	1	2	3
y	1	2	3	4	5

It’s easy to see the relationship- each value of y is bigger than the value of x by 2. Thus, is the function that represents this set of values. That’s all you really need to know in order to make functions.

There are also a few notations that are useful when dealing with functions.

The notation denotes the function of x. For example, is a function that inputs an x and returns x+2 as a y-value. If we’re looking for a particular value of a function, we can plug in any number at x, for example 3, to get . For example, if , then . This also hold for , and many other functions.

Functions provide us with a convenient way to describe relationships between 2 different variables, and thus they can be applied to real-life situations with variables we designate as x and y. For example, let’s say I run at 3 mph. If we call y my distance at any time x, the relationship between my distance and time can be written as .

So, do you know what functions are and what are they good for? Prove it, by writing the Basic Functions test.