Writing an Equation to Describe a Table - Problem 2
To write a function to describe a table of values, first see if there are any patterns that show up right away. If not, then consider the slope-intercept form of a line y=mx+b, where m = slope and b = y-intercept. The y-intercept is the point where the line crosses the y-axis. At that point, the coordinate points would have a x value of 0. Therefore, in the table of values, if there is a pair of values in which the x is 0, then that ordered pair represents the point where the graph of the function crosses the y-axis, or the y-intercept, which is "b". Next, to find the slope, use the slope formula and plug in the values from two ordered pairs from the table of values as the x1, y1 and x2, y2 values. Since slope is essentially the change in the x-value over the change in the y-value, you can also find the slope by determining the unit of change for the x and y. Once you find the value of the slope, plug it into the slope-intercept form of the line. The result is the function that describes the relationship in the given table of values. Make sure it is equal to f(x).
This problem is a little bit challenging because when you're looking at these number of boxes you can see it's decreasing. I start out with 1, 2, 3, 4, times 4. I start with 16 boxes then I come down to only 12 and then 8 and then 4. So it's a little bit different from some of the problems that you might have seen because these numbers are decreasing instead of increasing. The pictures are getting smaller.
So what I want to do is look for some kind of patterns, see if I can see anything that I can recognize as a mathematical pattern that I'm familiar with. Like for example in order to help me organize this, I'm going to make a table where I have picture number and then number of boxes and that's going to help me look for patterns. Like my picture numbers are 0, 1, 2, 3 and 4, 0, 1, 2, 3 and 4 and then if you count up how many boxes there are at 16, then we have 12, then we have 8 and then 4 and then this fourth picture has zero boxes. It's kind of weird we'll deal with it when we get to it.
Okay so that's just another way of representing the same information and I need to find an equation that describes this. So I'm going to start by writing an equation in words and I'll go and change it into mathematical symbols in a second. Number of boxes is going to be equal to something.
Well one thing I notice is that I'm starting with 16, my zero group started with 16 from there it's decreasing. So I'm going to say 16 take away something I don't know what's going to go there yet. Then let's look at how much it's going down by. Each time I'm subtracting 4 boxes, like I used to have four rows of four or four columns of four now I only have three columns of four, that's the one that got taken away.
Here again I'm taking away a column of 4 boxes from the previous picture. What's happening is I'm subtracting 4 every time. You can see it here, take away 4, take away 4 and on and on and on. So the way this pattern would be written would be -4 times the picture number, picture number. That's how I would write it in words or if I wanted to use mathematical symbols I would 16 take away 4x that rule or that equation describes how many boxes there are in each picture number.
So again this one's kind of weird because instead of getting bigger the pictures are getting smaller but you can still use your pattern techniques. Look for where you're starting from and then look at how much you're changing by. We started with 16 and then we're decreasing by four each time.