Answer:
a. zeros are: -3, 1, and 4
b. f(x) = 82(x - 1)(x + 3)(x - 4)
Step by Step:
a. You can think of it two ways:
1. (these are the x coordinates for the x-intercepts, or (-3, 0) (1, 0) and (4,0))
2. If you take the function of the graph and substitute/input any one of these 3 "zeros" or "intercepts" or "solutions" for the x variable, they will produce an output of 0.
b. To turn this into a function, we must write it in intercept form:
y = a(x-b)(x-c)
Since this is a cubic function, we must have three factors rather than two:
y = a(x - b)(x - c)(x - d)
Each zero we have found previously must substitute for the equation, and thus turned into factors:
-3, 1, and 4 are substituted for:
(x + 3), (x - 1), (x - 4)
Now, we have:
y = a(x+3)(x-1)(x-4)
To solve for a, find a point on the graph and substitute its x and y coordinate, and solve:
(.5, 5) is a point on the graph of the function, where x = .5 and y = 5. Using this point, we have:
5 = a(.5 + 3)(.5 - 1)(.5 - 4)
a (3.5)(-.5)(-3.5) = 5
6.125a = 5
a = 5/6.125
a ≈ .82
f(x) (or whatever the function is called) = .82(x - 1)(x + 3)(x - 4)