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Problem 632

# Graph the function $$\mathrm{y}=\mathrm{x}^{3}-9 \mathrm{x}$$.

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In summary: - x-intercepts: $$-3, 0, 3$$ - y-intercept: $$0$$ - critical points: $$-\sqrt{3}, \sqrt{3}$$ - end behavior: $$y \rightarrow -\infty$$ as $$x \rightarrow -\infty$$ and $$y \rightarrow +\infty$$ as $$x \rightarrow +\infty$$ With these features, sketch the graph of the function $$y = x^3 - 9x$$.
See the step by step solution

## Step 1: Find the x and y-intercepts

To find the x-intercepts, we need to set $$y = 0$$ and solve for $$x$$. So, $$0 = x^3 - 9x$$. To solve this equation, we can factor out an x: $$0 = x(x^2 - 9)$$. Now we have two factors: $$x$$ and $$(x^2 - 9)$$. Setting both equal to 0, we get $$x = 0$$ and $$x^2 - 9 = 0$$. The latter equation is a difference of squares and can be factored further into $$(x+3)(x-3)$$. Setting both factors equal to 0, we find two more x-intercepts: $$x=-3$$ and $$x=3$$. To find the y-intercept, we set $$x = 0$$ in the original equation. Thus, $$y = (0)^3 - 9(0) = 0$$. The intercepts are as follows: - x-intercepts: $$-3, 0, 3$$ - y-intercept: $$0$$

## Step 2: Find the critical points

Critical points occur when the first derivative is equal to 0 or does not exist. We will find the first derivative of the function, set it equal to 0, and solve for $$x$$: $$f'(x) = \frac{d}{dx}(x^3 - 9x)$$ Applying the power rule, we get: $$f'(x) = 3x^2 - 9$$ Now we set $$f'(x) = 0$$ and solve for $$x$$: $$3x^2 - 9 = 0$$ Divide by 3: $$x^2 - 3 = 0$$ Now, add 3 to both sides: $$x^2 = 3$$ Finally, take the square root of both sides: $$x = \pm\sqrt{3}$$ So, critical points occur when $$x = -\sqrt{3}, \sqrt{3}$$.

## Step 3: Analyze the end behavior

When analyzing the end behavior of a polynomial function, we look at the power of the leading term. In this case, the leading term is $$x^3$$, which is an odd power. As a result, the end behavior is such that as $$x$$ goes to negative infinity ($$-\infty$$), $$y$$ goes to negative infinity ($$-\infty$$), and as $$x$$ goes to positive infinity ($$+\infty$$), $$y$$ goes to positive infinity ($$+\infty$$). In summary: - x-intercepts: $$-3, 0, 3$$ - y-intercept: $$0$$ - critical points: $$-\sqrt{3}, \sqrt{3}$$ - end behavior: $$y \rightarrow -\infty$$ as $$x \rightarrow -\infty$$ and $$y \rightarrow +\infty$$ as $$x \rightarrow +\infty$$ With those features, we can sketch the graph of the function $$y = x^3 - 9x$$. It would start at the bottom left corner, cross the x-axis at $$-3$$, reach the first critical point at $$-\sqrt{3}$$, touch the x-axis at the origin, reach the second critical point at $$\sqrt{3}$$, cross the x-axis at $$3$$, and finally expand towards the top right corner.

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