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Writing an Equation to Describe a Table - Problem 3
In a table of values with points from a function, if there are no patterns that describe the relationship of the ordered pairs that stick out, use the slope-intercept form of a line to help you write the function. Remember that slope intercept form is 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, the slope can also be calculated by finding 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 is a problem where I'm given a series of pictures and I'm asked to write an equation to describe how many small triangles there would be and then the second part I have to do is figure out how many small triangles will the ninth picture have. And I'll tell you guys I just had to draw these, they're not easy to draw. I do not want to draw out to the ninth picture. So hopefully we can get this using a method that doesn't require drawing all these triangles.
Okay, first thing I want to do is try to organize my information, I'm going to try to set it up in a way that's a little different from this because it's hard for me to look at this and be able to tell right away how to approach this problem.
So I'm going to set up picture number and then number of triangles in a table and then my picture numbers are 0, 1, 2, 3, 4, 0, 1, 2, 3, 4 and then I have to count how many triangles there are. Zero has zero, one has one, that's not so bad, two has 1, 2, 3, 4 1, 2, 3, 4, 5, 6, 7, 8, 9, and then 16. So what I want you guys to start thinking about, is what pattern starts to like pop up in your brain. Hopefully when you see this group of numbers, bells and whistles go off in your brain. This is something that's really important in Math. Zero, one, four, nine, 16 those are what we call perfect square numbers.
I'm going to write that up here, perfect squares and what do you mean squares, those are triangles, I know what you mean. When talking about perfect squares what that means in Math is you're taking your picture number times-ing it by itself to get your number of triangles. Like 3 times 3 give you 9 that's called squaring and you write it like this 3² equals 9. So these groups of numbers zero, one, four, nine, 16 those are called perfect squares because each one has the square root that's the picture number.
So let's go back and answer the question, write and equation to describe the number of small triangles in the pattern. Okay well what I did was I took my picture number and I timesed it by itself. Picture number, we're going to write it as squared, that's what that little tiny two means that's equal to the number of triangles, number of triangles. If you wanted to write it using only Math you would write, y equals x² which means x times x, that's your equation. That describes how many triangles there are in each picture. Now for part B we're not going to have to draw the triangles, thank goodness we can just say how many will the ninth picture have. Okay I have to do my picture number squared or times itself that's going to give me 81.
That's the cool thing about Math you guys, it's all about shortcuts I could have drawn out all these gazillion 81 triangles it would have taken me all day or I can just use this rule finding the pattern helps me find the rule so I can just jump the answer. One thing that's really helpful in Math is finding the shortcuts and if you guys can start recognizing patterns like these perfect square numbers you're are going to be well on your way.