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# I NEED HELP

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Graph the function.

f(x)=-3x3-6x2+3x+6

Name the x-intercepts in the graph of the function.

Hello, Thank you for checking this question out. Would you please help me :) Step by step would be great

Aug 16, 2023

#1
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To graph the function $$f(x) = -3x^3 - 6x^2 + 3x + 6$$ and identify its x-intercepts, we can follow these steps:

1. Find the x-intercepts by setting $$f(x)$$ to 0 and solving for $$x$$:

$f(x) = -3x^3 - 6x^2 + 3x + 6 = 0$

2. Once we find the x-intercepts, we can plot them on the graph.

Let's proceed with these steps:

Step 1: Finding x-intercepts

To find the x-intercepts, we'll solve the equation $$-3x^3 - 6x^2 + 3x + 6 = 0$$. This equation can be factored to some extent:

$$-3x^3 - 6x^2 + 3x + 6 = -3(x^3 + 2x^2 - x - 2)$$

Now, let's use a tool like a graphing calculator or a computer algebra system to find the roots of the polynomial $$x^3 + 2x^2 - x - 2$$. The roots are approximately -1, 1, and 2.

So, the x-intercepts are: $$x = -1$$, $$x = 1$$, and $$x = 2$$.

Step 2: Graphing the function

Now that we have the x-intercepts, let's proceed to graph the function. Here's a rough sketch of the graph:


^
|                 *
|               *
|            *
|          *
|       *
|      *           * *
|    *          *
| *            *
|*___________*______________>
-2    -1   1   2


In this graph, the x-intercepts are marked with asterisks. The function $$f(x)$$ approaches negative infinity as $$x$$ moves towards negative infinity and approaches positive infinity as $$x$$ moves towards positive infinity. The graph shows a general shape of the cubic polynomial, with its behavior around the x-intercepts.

Please note that this is a rough sketch and not to scale. For a more accurate and detailed graph, you can use graphing software or a graphing calculator.

Aug 16, 2023
#2
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Usually, finding the x-intercepts of a cubic function is quite challenging, but some observation reveals we can use factoring to find the x-intercepts of this particular cubic function $$f(x) = -3x^3 - 6x^2 + 3x + 6$$ by setting the function equal to 0 and then solving for the individual x-values.

$$-3x^3 - 6x^2 + 3x + 6 = 0 \\ -3(x^3 + 2x^2 -x - 2 = 0 \\ -3[x^2(x + 2) - 1(x + 2)] = 0 \\ -3(x^2 - 1)(x + 2) = 0 \\ -3(x - 1)(x + 1)(x + 2) = 0 \\ x=1 \text{ or } x = -1 \text{ or } x = -2$$

These x-values indicate the x-coordinate of the x-intercept. The y-coordinate of the x-intercept is always 0. Therefore, the x-intercepts are $$(1, 0), (-1, 0), \text{ and } (-2, 0)$$.

With this information of the x-intercept and the negative nature of the leading coefficient, you should be able to make a reasonably accurate graph of this cubic function, but below is one for reference.

Aug 17, 2023
#3
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Okay, I appreciate you so much. Thank you The3Mathketeers :))

thebestchesscat  Aug 17, 2023