Sketching the Graph of f(x) = 3x^{3} - 4x^{2} - 5x + 2
 
 


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.

Sketching the Graph of f(x) = 3x^{3} - 4x^{2} - 5x + 2

Explore how to sketch the graph of the cubic function f(x) = 3x^3 - 4x^2 - 5x + 2. This detailed guide covers key concepts like derivatives, critical points, and inflection points, providing a step-by-step approach to graphing polynomial functions. Suitable for advanced high school and college-level students.

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