Building quadratic functions from geometric patterns
About this video
The video uses the rule f(n) = n^2 + 2 to find the number of squares in the first three steps of a growing pattern (3, 6 and 11). It then picks the pattern picture that matches those counts, choice B.
You'll be able to
- evaluate a quadratic rule like f(n) = n^2 + 2 for step numbers.
- match a growing pattern of squares to a quadratic rule.
Key words
- step number
- Which stage of a growing pattern you are looking at: step 1, step 2 and so on.
- f(n)
- The output of the function for input n; here, the number of squares in step n.
Transcript
- [Instructor] Let n be the step number in a growing pattern. Here is a function that gives the total number of squares in step n. f of n is equal to n squared plus two. Which pattern matches this rule for the first three steps? All right, pause the video, like always.
Have a go at it. All right, now let's do it together. So let's just think about this. So step n, actually, let me just think of it this way. I'll make a little table here. So we're going to think about the step here, or let me just call that n, n here. And then we're gonna say what f of n is.
So at the first step, we're going to have f of n is going to be equal to one squared plus two, which is equal to three. At the second step, f of two is two squared, which is four, plus two, which is six. At the third step, we're gonna have three squared, which is nine, plus two, which is 11.
So which of these meets that? So on the first step, we should have three. So immediately, I can rule out A and C. But let's just see if it keeps up that way. So this checks out, step one. Step two, I should have six. And indeed I have one, two, three, four, five, six squares here.
That checks out. Step three, I should have 11, and I have nine over there, plus one, plus one. So that is indeed 11, so I like choice B.
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