Answer:
c. f(x)=x(x+2)(x-1)(x-4)
Explanation:
Where it says "as x goes to negative infinity" then "y goes to infinity" (that's the part with the infinity symbols) that means the graph is going up at it's left end. This curve is a quartic (4th degree) which means its left and right ends are kind of parabola-ish, but the middle is not the neat u-ish, v-ish shape of a parabola; it's more like a wonky, noodle-ish wavy affair. Anyway, LIKE a parabola when the beginning of the equation is positive, the two ends point up. That's what's happening here and so we can eliminate b. and d. as potential answers.
Since -2, 0, 1, 4 are zeros (which are solutions...and also x-intercepts) we can find the factors of the function.
If x = -2
ADD 2 to both sides.
x + 2 = 0
This means (x+2) is a factor.
This is enough info to select answer c. but let's verify the other factors.
If x = 1
SUBTRACT 1 from both sides.
x - 1 = 0
Thus means (x - 1) is a factor.
If x = 4
SUBTRACT 4 from both sides.
x - 4 = 0
(x - 4) is a factor.
You can see c. has all these factors as well as x, because x=0 already, so x is a factor too.
I think of this as a working backwards problem, bc usually you have to factor and solve. This one, you have solutions (which are zeros and x-intercepts) and work backwards to find factors and multiply them together to find the function.