To sketch the graph of the function $f(x) = x^3 - 2x^2 + 4x + 1$ using Key Idea 3.5.1 (likely related to analyzing critical points, inflection points, and behavior at infinity), let’s go through these steps:

Step 1: Identify the Function and its Derivatives

1. Given function: $f(x) = x^3 - 2x^2 + 4x + 1$.
2. First derivative $f'(x)$: To find critical points (where $f'(x) = 0$), we’ll take the derivative: $f'(x) = 3x^2 - 4x + 4$
3. Second derivative $f''(x)$: To find inflection points, we take the derivative of $f'(x)$: $f''(x) = 6x - 4$

Step 2: Find Critical Points (Using $f'(x) = 0$)

Set $f'(x) = 0$ to find values of $x$ where the function might have a maximum, minimum, or saddle point. $3x^2 - 4x + 4 = 0$ Using the quadratic formula: $x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 3 \cdot 4}}{2 \cdot 3}$ $x = \frac{4 \pm \sqrt{16 - 48}}{6} = \frac{4 \pm \sqrt{-32}}{6}$ This results in imaginary values, so there are no real critical points where $f'(x) = 0$. Thus, $f(x)$ has no local maximum or minimum points.

Step 3: Find Inflection Points (Using $f''(x) = 0$)

Set $f''(x) = 0$ to find potential inflection points: $6x - 4 = 0$ $x = \frac{4}{6} = \frac{2}{3}$ When $x = \frac{2}{3}$, there is a potential inflection point.

Step 4: Analyze the End Behavior

Since $f(x) = x^3 - 2x^2 + 4x + 1$ is a cubic function, we know that as $x \to \infty$, $f(x) \to \infty$, and as $x \to -\infty$, $f(x) \to -\infty$.

Step 5: Sketch the Graph

• End behavior: Starts from $-\infty$ as $x \to -\infty$ and rises to $+\infty$ as $x \to \infty$.
• Inflection point: At $x = \frac{2}{3}$, the curve changes concavity.
• No local maxima or minima because the critical points are complex.

Would you like a detailed sketch or have further questions on any of these steps?

Here are some related questions to further explore this topic:

1. How do you determine the concavity of a function at a given point?
2. What is the importance of the first derivative in graphing functions?
3. How can you verify if a function has a local maximum or minimum using derivatives?
4. Why does a cubic function exhibit end behavior going to $+\infty$ and $-\infty$?
5. How does an inflection point affect the shape of the graph?

Tip: For sketching, always check inflection points and end behavior to capture the main shape of the curve.