Question

answer parts (a)-(e) for the function shown below. f(x)=x³ + 2x² - x -
e. use the graphing tool to graph the function. if necessary, find a few additional points and graph the function. use the maximum number of turning points to check whether it is drawn correctly. click to enlarge graph

Explanation:

Step1: Find the y - intercept

Set \(x = 0\) in \(f(x)=x^{3}+2x^{2}-x\). Then \(f(0)=0^{3}+2\times0^{2}-0 = 0\). So the y - intercept is \((0,0)\).

Step2: Find the x - intercepts

Set \(f(x)=0\), so \(x^{3}+2x^{2}-x=x(x^{2}+2x - 1)=0\). Using the quadratic formula for \(x^{2}+2x - 1 = 0\), where \(a = 1\), \(b = 2\), \(c=-1\), \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}=\frac{-2\pm\sqrt{4 + 4}}{2}=\frac{-2\pm2\sqrt{2}}{2}=-1\pm\sqrt{2}\). The x - intercepts are \(x = 0\), \(x=-1+\sqrt{2}\approx0.414\), \(x=-1 - \sqrt{2}\approx - 2.414\).

Step3: Determine the end - behavior

Since the leading term of \(f(x)=x^{3}+2x^{2}-x\) is \(x^{3}\) (with odd degree and positive leading coefficient), as \(x\to-\infty\), \(y\to-\infty\) and as \(x\to+\infty\), \(y\to+\infty\).

Step4: Find the turning points

The degree of the polynomial \(n = 3\), so the maximum number of turning points is \(n - 1=2\). We can find the derivative \(f'(x)=3x^{2}+4x - 1\). Set \(f'(x)=0\), then using the quadratic formula for \(3x^{2}+4x - 1 = 0\) (\(a = 3\), \(b = 4\), \(c=-1\)), \(x=\frac{-4\pm\sqrt{16+12}}{6}=\frac{-4\pm\sqrt{28}}{6}=\frac{-4\pm2\sqrt{7}}{6}=\frac{-2\pm\sqrt{7}}{3}\). \(x_1=\frac{-2+\sqrt{7}}{3}\approx0.215\), \(x_2=\frac{-2 - \sqrt{7}}{3}\approx - 1.549\).

Step5: Plot points and graph

We can find some additional points, for example, when \(x=-3\), \(f(-3)=(-3)^{3}+2\times(-3)^{2}-(-3)=-27 + 18+3=-6\); when \(x = 1\), \(f(1)=1^{3}+2\times1^{2}-1=2\); when \(x = 2\), \(f(2)=2^{3}+2\times2^{2}-2=8 + 8-2 = 14\). Plot the x - intercepts, y - intercept, additional points, consider the end - behavior and turning points to graph the function.

Since this is a graphing task and we are not asked for a specific numerical answer, we have provided the steps to graph the function. If you were to graph it on a graphing utility or by hand - plotting points, you would follow the above steps.

Answer:

Step1: Find the y - intercept

Set \(x = 0\) in \(f(x)=x^{3}+2x^{2}-x\). Then \(f(0)=0^{3}+2\times0^{2}-0 = 0\). So the y - intercept is \((0,0)\).

Step2: Find the x - intercepts

Set \(f(x)=0\), so \(x^{3}+2x^{2}-x=x(x^{2}+2x - 1)=0\). Using the quadratic formula for \(x^{2}+2x - 1 = 0\), where \(a = 1\), \(b = 2\), \(c=-1\), \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}=\frac{-2\pm\sqrt{4 + 4}}{2}=\frac{-2\pm2\sqrt{2}}{2}=-1\pm\sqrt{2}\). The x - intercepts are \(x = 0\), \(x=-1+\sqrt{2}\approx0.414\), \(x=-1 - \sqrt{2}\approx - 2.414\).

Step3: Determine the end - behavior

Since the leading term of \(f(x)=x^{3}+2x^{2}-x\) is \(x^{3}\) (with odd degree and positive leading coefficient), as \(x\to-\infty\), \(y\to-\infty\) and as \(x\to+\infty\), \(y\to+\infty\).

Step4: Find the turning points

The degree of the polynomial \(n = 3\), so the maximum number of turning points is \(n - 1=2\). We can find the derivative \(f'(x)=3x^{2}+4x - 1\). Set \(f'(x)=0\), then using the quadratic formula for \(3x^{2}+4x - 1 = 0\) (\(a = 3\), \(b = 4\), \(c=-1\)), \(x=\frac{-4\pm\sqrt{16+12}}{6}=\frac{-4\pm\sqrt{28}}{6}=\frac{-4\pm2\sqrt{7}}{6}=\frac{-2\pm\sqrt{7}}{3}\). \(x_1=\frac{-2+\sqrt{7}}{3}\approx0.215\), \(x_2=\frac{-2 - \sqrt{7}}{3}\approx - 1.549\).

Step5: Plot points and graph

We can find some additional points, for example, when \(x=-3\), \(f(-3)=(-3)^{3}+2\times(-3)^{2}-(-3)=-27 + 18+3=-6\); when \(x = 1\), \(f(1)=1^{3}+2\times1^{2}-1=2\); when \(x = 2\), \(f(2)=2^{3}+2\times2^{2}-2=8 + 8-2 = 14\). Plot the x - intercepts, y - intercept, additional points, consider the end - behavior and turning points to graph the function.

Since this is a graphing task and we are not asked for a specific numerical answer, we have provided the steps to graph the function. If you were to graph it on a graphing utility or by hand - plotting points, you would follow the above steps.