To graph the function $g(x) = -x^5 - 3x^4 + 40x^3$, follow these steps:

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Step 1: Analyze the Function

1. Degree and Leading Coefficient:

   • The degree of the polynomial is 5 (odd degree).
   • The leading coefficient is $-1$, so the graph will start in the top-left quadrant and end in the bottom-right quadrant.

2. Critical Points:

   • Take the derivative of $g(x)$ to find where the slope of the graph is zero (critical points).
   • $g'(x) = -5x^4 - 12x^3 + 120x^2$.

3. Factor the Derivative: Factorize $g'(x)$ to find critical points: $g'(x) = -5x^2(x^2 + 2.4x - 24).$ Solve $x^2 + 2.4x - 24 = 0$ using the quadratic formula: $x = \frac{-2.4 \pm \sqrt{2.4^2 - 4(1)(-24)}}{2(1)}.$

4. Behavior Near Roots:

   • Solve $g(x) = 0$ for $x$.
   • The roots are where the function intersects the x-axis. Use factoring or numerical methods to estimate them.

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Step 2: Create a Table of Values

Choose points around critical points and intercepts. For example:

• Plug values like $x = -3, -2, -1, 0, 1, 2, 3$ into $g(x)$.
• Calculate the corresponding $y$-values.

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Step 3: Plot and Sketch

1. Plot the roots (where $g(x) = 0$).
2. Plot the critical points (local maxima and minima).
3. Plot additional points from the table to get an accurate curve.
4. Draw the curve, ensuring it matches the behavior for odd-degree polynomials (falling right end, rising left end).

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Step 4: Use Technology

If manual plotting is challenging, graphing tools like Desmos or a graphing calculator can help.

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Would you like me to work out all roots and critical points explicitly, or show how this graph looks? Let me know!

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Follow-Up Questions:

1. Do you understand how to derive $g'(x)$ to find critical points?
2. Should we calculate the exact turning points step-by-step?
3. Would you like me to help create a table of values for plotting?
4. Are you familiar with using graphing tools or calculators for polynomials?
5. Would you like clarification on how the degree affects the end behavior of the graph?

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Tip:

Always analyze the end behavior and symmetry of a polynomial function to sketch a more accurate graph.