Please note: This explanation may appear to be incomplete because Brainly software falsely accuses me of using swear words or a URL. I have deleted some of my answer to get Brainly to accept part of it.
The y-intercept is found by setting x = 0: f(0) = (0)^3 − 11(0)^2 + 36(0) − 36. We get f(0) = -36. The y-intercept is thus (0, -36).
The x-intercepts are found from the roots/zeros of x^3 − 11x^2 + 36x − 36.
Synthetic division is very helpful in finding the roots of polynomials. We list possible factors of the constant term -36 and test each in
x^3 − 11x^2 + 36x − 36 until we find one or more factors of -36 for which the remainder is zero. -36 has the possible factors ±1, ±2, ±3, ±4, ±6, ±12, ±18, ±36.
Let's try -4. Is this a root of x^3 − 11x^2 + 36x − 36?
-4 / 1 -11 36 -36
4 28 256
------------------------------ No, -4 is not a root of x^3 − 11x^2 + 36x − 36
1 -7 64 220 because the remainder is 220, not 0.
Eliminate the possibility x = -4. Continue in the same way, testing each possible factor of -36 as a zero remainder. Try x = 2. We get:
2 / 1 -11 36 -36
2 -18 36
---------------------------
1 -9 18 0 Since the remainder is 0, we know that 2
is a root of x^3 − 11x^2 + 36x − 36.
Similarly, test x = 3: Using the coefficients 1 -11 36 -36, we obtain a zero remainder and thus can conclude that 3 is a root or zero of the original equation.
Thus, the roots/zeros of the original equation are {2, 3, 6}
and the corresponding factors of this equation are thus
{(x - 2), (x - 3), (x - 6). The x-intercepts are {(0, 2), (0, 3), (0, 6)} This function has no negative root/roots. Thus, the entire graph lies to the right of the y-axis. The curve increases/rises between x = 0 and x = 2 and crosses the x-axis at (0, 2). Then it decreases/drops between x = 2 and x = 3. Finally, the curve increases/rises again to the right of x = 6, always increasing.
