Answers:
- x-intercepts: -9, 0, 6
- y-intercept: 0
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Explanation:
To find the x intercept(s), we replace f(x) with 0 and solve for x.
[tex]f(\text{x}) = -\text{x}^3 - 3\text{x}^2 + 54\text{x}\\\\0 = -\text{x}^3 - 3\text{x}^2 + 54\text{x}\\\\-\text{x}^3 - 3\text{x}^2 + 54\text{x} = 0\\\\-\text{x}(\text{x}^2 + 3\text{x} - 54) = 0\\\\-\text{x}(\text{x} + 9)(\text{x} -6) = 0\\\\[/tex]
Set each factor equal to x and solve those equations.
[tex]-\text{x} = 0 \ \text{ or } \ \text{x}+9 = 0 \ \text{ or } \ \text{x}-6 = 0\\\\\text{x} = 0/(-1) \ \text{ or } \ \text{x} = 0-9 \ \text{ or } \ \text{x} = 0+6\\\\\text{x} = 0 \ \text{ or } \ \text{x} = -9 \ \text{ or } \ \text{x} = 6\\\\[/tex]
Therefore the x intercepts are 0, -9 and 6 in any order of your choosing.
It probably makes more sense to have the numbers in increasing order. So I would write the answer as -9, 0, 6 for the x intercepts portion.
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To find the y intercept, we replace every copy of x with 0 and simplify
[tex]f(\text{x}) = -\text{x}^3 - 3\text{x}^2 + 54\text{x}\\\\f(0) = -(0)^3 - 3(0)^2 + 54(0)\\\\f(0) = 0\\\\[/tex]
Everything goes to zero because the variable x is part of each term (and we replaced said variable with zero).
Therefore, the y intercept is 0. This also is one of the x intercepts mentioned earlier. In other words, this y intercept is at the origin (0,0).
At most one y-intercept is allowed if we wanted a function.
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The graph is shown below to confirm the answers.
I used GeoGebra but you could use Desmos or any other similar software, or something like a TI83.
