Chapter 21: Chapter 21

Problem 632

Graph the function \(\mathrm{y}=\mathrm{x}^{3}-9 \mathrm{x}\).

Short Answer

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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 \(y = x^3 - 9x\).

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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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Chapter 21: Chapter 21

Graph the function \(\mathrm{y}=\mathrm{x}^{3}-9 \mathrm{x}\).

Short Answer

Step by step solution

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Step 1: Find the x and y-intercepts

Step 2: Find the critical points

Step 3: Analyze the end behavior

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