This week’s Math Journey builds on the material in The Function Machine. If you have not yet read that journey, I suggest you do so now.
In The Function Machine we discussed why graphing a function is possible at all on a conceptual level – essentially, since every x value of a function has a corresponding y value, we can plot those corresponding values as an ordered pair on a coordinate plane. Plot enough pairs and a pattern begins to emerge; we join the points into a continuous line as an indication that there are actually an infinite number of pairs when you account for all real numbers as possible x values.
But plotting point after point is a tedious and time-consuming process. Wouldn’t it be great if there was a quick way to tell what the graph was going to look like, and to be able to sketch it after plotting just a few carefully-chosen points?
Well, there is! Mathematicians look for an assortment of clues that help to determine the shape of a function’s graph from the equation itself – and it’s those clues that we’ll be talking about today. They come in four basic flavors: the power, the sign, the co-efficient, and the constant.
Let’s start with our old standby from the previous journey: y = x + 4. When we talk about “power” in this context, we’re referring specifically to the highest exponent on an x value. The highest power in this problem is one; there are no exponents so the x is simply raised to the first power. This means that for every value of y, there is exactly one corresponding value of x. If x is 1, y is 5. If x is 2, y is 6, and so on. For every given increase in x, there is a proportional increase in y, in this case it’s 1 to 1. And that means that this graph is a straight line. Easy enough, right?
Well, let’s throw a bit of a wrench into the works here, shall we? Your new function is y = x^2. Now, if I turned the machine around backwards and told you that y was 4, what would you give me for x? You might give me 2, right? 2 squared is 4. But hang on, there’s more than one thing you can square to get 4.
Not seeing it?
How about negative 2? When you square a negative number it goes positive, right? So your x value could just as easily have been – 2 as positive 2. And the same thing would have been true for any value of y, right – the corresponding x value could be either the square root of y or the negative square root of y. So in this case, there is more than one corresponding x value for any given value of y – in fact there’s exactly 2 corresponding x values for each y (with the exception of 0, of course). That means that this graph is NOT a straight line.
Turns out, it’s actually a parabola. All functions with x^2 as their highest power (known as quadratic functions) graph out as parabolas. The specific parameters of each parabola are determined by the other categories of clues, but the power tells us that this graph will be some kind of parabola. In the same way, the powers of higher-power functions also tell us the type of shape they will graph; third power functions (ones with a cube as their highest power) will form hyperbolas, and so on. This holds true with functions that include radicals as well; the type of power indicates the rough shape of the graph.
Let’s take our quadratic function of y = x^2. When you plot some points it becomes clear that this is a parabola opening upwards; the larger the x values become, the exponentially larger the y value becomes. But what if I made one slight change to this equation?
Y = – (x^2)
Now I’m asking you, essentially, to take each of those y values and invert it. If x is 2 (or negative 2), y would now be negative 4. This holds true for every value of y, so if you plot a few of those sets of points it quickly appears that you’ve just flipped the parabola upside down. And indeed, the sign on the highest-power x value dictates which direction the graph will be facing (at least in terms of up-and-down; the side-to-side graphs are usually dictated by higher powers in the first place or by radicals or other more complex types of functions). If we were dealing with a straight line, the negative sign would indicate that the line travels downward as it moves to the right, rather than upward. Y = -x, for example, is a line with a negative slope, which means it moves down and to the right rather than up and to the right as y = x does. If you graphed both of those line functions, they’d come out to be mirror images of each other. So the sign on the highest-power x value dictates direction.
When we talk about a co-efficient in math, we’re generally referring to the number that is multiplied by a variable. Take, for example, the function y = 3(x^2). How would this differ from our original y = x^2?
Well, let’s follow the problem through. With a co-efficient, each time we get the square we’ll need to multiply it by 3 before it becomes the y value. This will mean that each y value is quite a bit larger than the y value in our original problem. The curve will be quite a bit steeper, since using 2 for x will give us 12 for y instead of 4. So with a co-efficient above 1, the graph will show up steeper/skinnier/more closed. With a co-efficient that is a fraction, however, the graph will show up shallower or more open. Think about y = (1/3)(x^2). With 2 for x, you’d now end up with 4/3 for y; even less than with the original problem. So the co-efficient tells us how steep or sharp the progression of the curve is. Higher numbers mean sharper curves, while smaller fractions mean more gentle progressions.
The constant is my favorite clue. A constant is a number that does not involve a variable. In our original y = x + 4, that +4 is the constant. That constant is the y-intercept – the value at which x is 0. If x were 0, all terms with x’s in them would become zeros and all you’d have left would be the constant. So with a quick look at the constant you can figure out one of your points with no work at all. But here’s the really fun part. Since it doesn’t involve a variable, the constant doesn’t actually change the shape of the curve itself. What it does do is move it around the plane. Take a look at y = x^2 versus y = x^2 + 4. That +4 on the end simply means that every y value you normally would have gotten is now 4 places higher on the graph. The whole curve has been lifted up four places on the graph. If it were a negative 4 – you guessed it – it would have moved down four places.
So the natural next question is: what if you want to move it right or left on the plane? Well, that involves getting a second co-efficient into play. Let’s change our equation to x^2 + 2x + 4. That 2x will shift the graph horizontally – but it’s a little bit more complicated than you might think. The signs here are actually reversed – adding 2x moves the graph to the left, and subtracting it moves the graph to the right. Also, it’s not a one-to-one ratio; in fact the ratio varies depending on the equation itself. Remember, too, that the constant is still the y-intercept, so if you get sideways transposition involved the center won’t necessarily be cleanly at an easily-discernible value anymore; but the curve will still cross the y-axis at 4. Combining those two pieces of information, along with the power, sign, and leading co-efficient to tell you the shape of the curve, will get you well on your way to knowing what the graph looks like.
Remember back at the beginning when I told you that using these clues would allow you to plot just a few points and sketch the graph more quickly? Well, here’s how we put it all together. Let’s take a new equation:
y = 3(x^2) + 5x – 2
What can we tell about the graph from the clues presented here?
First, the power. This is a quadratic function, which means we’re dealing with a parabola. The leading sign is positive, so it’ll open upward. The leading co-efficient is 3, which is greater than 1, so it’ll be a sharper, steeper curve, 3 times steeper than the basic parabola. We’re adding 5x, so the graph will be transposed to the left, and the y-intercept is at – 2. We’d still need to work out and plot a couple of points (personally, I’d factor the quadratic to find the x-intercepts and work from there – more on that next time), but now we have a better idea of what the graph would look like – and we can see all of that just from the equation alone!