To sketch the graph of the function $f(x) = 3x^3 - 4x^2 - 5x + 2$, let's go through a step-by-step process.

1. Determine the Behavior of the Function (End Behavior):

• Since this is a cubic polynomial ($x^3$ term is the highest power), the end behavior will follow the pattern of a cubic function.
• The coefficient of $x^3$ is positive (3), so as $x \to \infty$, $f(x) \to \infty$ and as $x \to -\infty$, $f(x) \to -\infty$.

2. Find the First Derivative (for Critical Points):

• The first derivative, $f'(x)$, will help identify any critical points (local maxima, minima).

• $f(x) = 3x^3 - 4x^2 - 5x + 2$

• Differentiating with respect to $x$: $f'(x) = 9x^2 - 8x - 5$

• Set $f'(x) = 0$ to find critical points: $9x^2 - 8x - 5 = 0$

• Solving this quadratic equation using the quadratic formula: $x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \cdot 9 \cdot (-5)}}{2 \cdot 9}$ $x = \frac{8 \pm \sqrt{64 + 180}}{18}$ $x = \frac{8 \pm \sqrt{244}}{18} \approx \frac{8 \pm 15.62}{18}$ $x \approx 1.3 \quad \text{and} \quad x \approx -0.42$

These are approximate values for the critical points.

3. Find the Second Derivative (for Concavity):

• The second derivative, $f''(x)$, helps determine the concavity of the function.

• Differentiating $f'(x)$: $f''(x) = 18x - 8$

• Set $f''(x) = 0$ to find potential inflection points: $18x - 8 = 0$ $x = \frac{8}{18} = \frac{4}{9} \approx 0.44$

At $x \approx 0.44$, there is a potential inflection point where the concavity changes.

4. Evaluate Key Points (Including Intercepts):

• Y-intercept: Set $x = 0$: $f(0) = 3(0)^3 - 4(0)^2 - 5(0) + 2 = 2$ So, the y-intercept is $(0, 2)$.

• X-intercepts: Solve $f(x) = 0$: Solving $3x^3 - 4x^2 - 5x + 2 = 0$ may require numerical or graphical methods as it does not factor nicely.

5. Plot and Sketch the Graph:

Using the information gathered:

• Plot the intercepts and mark critical points around $x \approx 1.3$ and $x \approx -0.42$.
• Mark the inflection point around $x \approx 0.44$.
• Show the end behavior: the graph moves up to infinity as $x \to \infty$ and down to negative infinity as $x \to -\infty$.

If you'd like a visual graph plotted from these steps or have questions on any step, feel free to ask!

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Follow-Up Questions:

1. What are critical points, and how are they identified on a graph?
2. How does concavity affect the shape of a function's graph?
3. Why is the end behavior important in understanding polynomial functions?
4. How can the quadratic formula help solve for critical points in a cubic function?
5. What is the significance of an inflection point on a graph?

Tip:

To sketch polynomials accurately, start by finding intercepts, critical points, and concavity. These elements together guide the overall shape of the graph.