Given the function $P(x)=x^{3}-1 x^{2}-12 x$, find its $y$-intercept is its $x$-intercepts are $x_{1}=$ , $x_{2}=$ and $x_{3}=$ with $x_{1}
Solution
Step 1 ：The y-intercept of a function is the point where the function crosses the y-axis. This occurs when x = 0. So, to find the y-intercept, we substitute x = 0 into the function and calculate the corresponding y value. In this case, the y-intercept is \(\boxed{0}\).
Step 2 ：The x-intercepts of a function are the points where the function crosses the x-axis. This occurs when y = 0. So, to find the x-intercepts, we solve the equation P(x) = 0. The solutions to this equation are the x-intercepts of the function. In this case, the x-intercepts are \(\boxed{-3}\), \(\boxed{0}\), and \(\boxed{4}\).
Step 3 ：As x approaches infinity, we need to consider the highest degree term in the function, as this will dominate the behavior of the function. In this case, the highest degree term is x^3, so as x approaches infinity, y will also approach infinity. Therefore, when x approaches infinity, y approaches \(\boxed{+ \infty}\).
Step 4 ：Similarly, as x approaches negative infinity, we again consider the highest degree term. Since the coefficient of x^3 is positive, as x approaches negative infinity, y will also approach negative infinity. Therefore, when x approaches negative infinity, y approaches \(\boxed{- \infty}\).
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Solution
Step 1 ：The y-intercept of a function is the point where the function crosses the y-axis. This occurs when x = 0. So, to find the y-intercept, we substitute x = 0 into the function and calculate the corresponding y value. In this case, the y-intercept is \(\boxed{0}\).
Step 2 ：The x-intercepts of a function are the points where the function crosses the x-axis. This occurs when y = 0. So, to find the x-intercepts, we solve the equation P(x) = 0. The solutions to this equation are the x-intercepts of the function. In this case, the x-intercepts are \(\boxed{-3}\), \(\boxed{0}\), and \(\boxed{4}\).
Step 3 ：As x approaches infinity, we need to consider the highest degree term in the function, as this will dominate the behavior of the function. In this case, the highest degree term is x^3, so as x approaches infinity, y will also approach infinity. Therefore, when x approaches infinity, y approaches \(\boxed{+ \infty}\).
Step 4 ：Similarly, as x approaches negative infinity, we again consider the highest degree term. Since the coefficient of x^3 is positive, as x approaches negative infinity, y will also approach negative infinity. Therefore, when x approaches negative infinity, y approaches \(\boxed{- \infty}\).
