Question

Medal! What are the apparent zeroes of the function graphed below.
a. (-0.7, -2.7)
b. (-10.2,10.2)
c. (-2,1,4)
d. (-4,-1,2)

Answer 1

What are the values of \(x\) where the graph crosses the x-axis? "zeros" of a function are just the values of \(x\) where \(f(x)=0\)

Answer 2

-2 and 4?

Answer 3

Doesn't it cross 3 times?

Answer 4

so 1 too?

Answer 5

look at the point x=1. if the curve is crossing the x-axis there, then yes, 1 too.

Answer 6

oh ok thanks!

Answer 7

easy, right?

Answer 8

from that you can also figure out the equation of the curve!

because \(f(x) = 0\) at \(x=-2,x=1, x = 4\), that means we can write
\[f(x) = a(x-(-2))(x-1)(x-4) = a(x+2)(x-1)(x-4)\]and then just set \(a\) so that we get the right value of \(f(x)\) at some point of our choosing (where \(f(x) \ne 0\))

Looks like the curve goes through the point (2,-8), so we plug in \(x=2\) and choose \(a\) so that \[-8 = a(2+2)(2-1)(2-4)\]\[-8 = a(4)(1)(-2)\]\[-8=-8a\]\[a=1\]

So our function that produced that graph was \[f(x) = 1(x+2)(x-1)(x-4) = x^3-3 x^2-6 x+8\]