Would this be worth showing to a math class?

Asked by LostInParadise (32183

) June 9th, 2021

After learning about inverse functions and quadratic equations, I thought students might enjoy seeing a connection between the two.

Suppose we are given the function f(x)= -x**2 + 6x + 10, and we want to find the axis of symmetry. Instead of completing the square we rewrite the function as f(x) = -x(6-x) + 10.

What do we get for f(6 – x)? We notice that 6-x is self-inverse and that f contains terms for both x and 6-x . For f(6-x), x gets mapped to 6 -x and 6-x gets mapped to x. f(6-x)= -(6-x)x + 10, which is the same as f(x), so f(6-x)=f(x). For every point (x,y) on the curve there will be a point (6-x,y) on the curve. The midpoint of (x,y) and (6-x,y) is (3,y), so the line x=3 is a line of symmetry.

This technique can be extended to all quadratic expressions and can be used for other types of functions, like f(x)=arctan(x)*arctan(6-x), which will also have a line of symmetry for x=3.

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4 Answers

That equation should have been f(x)=x**2–6x+10

LostInParadise (32183

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I just blacked out reading this, be nice to the kids. Haha!

KNOWITALL (29885

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Let me try another way. We have f(x) = -x(6-x)+10

Take any two numbers that add to 6, like 2 and 4.
x(6-x) in the case of x=2 gives 2×4=8, and in the case of x = 4 we get 4×2=8, the same value. For both x=2 and x=4, we get f(x)=-4×2+10 = -8+10 = 2, so we know that the graph of f(x) has the two points (2,2) and (4,2). The midpoint is (3,2).

Similarly, for any two numbers a and b that add to 6, we will get x(6-x) = ab and f(a)=f(b) = -ab+10 = y for some number y. That means that (a,y) and (b,y) are on the graph and the midpoint is (a+b/2,y) = (3,y), because a+b=6.

We keep getting midpoints of the form (3,y), so the line x=3 must cut the curve in two and is the axis of symmetry.

LostInParadise (32183

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