Answer :
To graph the function [tex]\( f(x) = x^3 + 9x^2 + 2x - 48 \)[/tex], follow these steps:
1. Identify the Type of Function:
- The given function is a cubic polynomial (degree 3).
2. Find the Critical Points:
- To find critical points, we need the first derivative [tex]\( f'(x) \)[/tex]:
[tex]\[
f'(x) = \frac{d}{dx}(x^3 + 9x^2 + 2x - 48) = 3x^2 + 18x + 2
\][/tex]
- Set the first derivative to zero to find critical points:
[tex]\[
3x^2 + 18x + 2 = 0
\][/tex]
- Solve the quadratic equation for [tex]\( x \)[/tex] using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]:
[tex]\[
x = \frac{-18 \pm \sqrt{18^2 - 4 \cdot 3 \cdot 2}}{2 \cdot 3} = \frac{-18 \pm \sqrt{324 - 24}}{6} = \frac{-18 \pm \sqrt{300}}{6} = \frac{-18 \pm 10\sqrt{3}}{6} = \frac{-9 \pm 5\sqrt{3}}{3}
\][/tex]
- Thus, the critical points are:
[tex]\[
x = -3 \pm \frac{5\sqrt{3}}{3}
\][/tex]
3. Evaluate the Function at Critical Points:
- Use these [tex]\( x \)[/tex] values to find the corresponding [tex]\( y \)[/tex] values:
[tex]\[
f\left(-3 + \frac{5\sqrt{3}}{3}\right) \quad \text{and} \quad f\left(-3 - \frac{5\sqrt{3}}{3}\right)
\][/tex]
- Substitute back in to find [tex]\( f(x) \)[/tex] at these points for exact values or numerical approximations.
4. Find Intercepts:
- Y-Intercept: Set [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = -48
\][/tex]
- X-Intercepts: For [tex]\( f(x) = 0 \)[/tex]:
[tex]\[
x^3 + 9x^2 + 2x - 48 = 0
\][/tex]
- Find the roots using methods like factoring, synthetic division, or numerical methods.
5. Analyze End Behavior:
- For [tex]\( x \rightarrow \infty \)[/tex] and [tex]\( x \rightarrow -\infty \)[/tex], cubic functions have:
[tex]\[
\text{As } x \rightarrow \infty, f(x) \rightarrow \infty
\][/tex]
[tex]\[
\text{As } x \rightarrow -\infty, f(x) \rightarrow -\infty
\][/tex]
6. Plot the Graph:
- Use the information from the steps above to sketch the graph.
- Mark the critical points, intercepts, and the general shape considering the end behavior.
By piecing this information together, you should be able to accurately sketch a graph of [tex]\( f(x) = x^3 + 9x^2 + 2x - 48 \)[/tex].
1. Identify the Type of Function:
- The given function is a cubic polynomial (degree 3).
2. Find the Critical Points:
- To find critical points, we need the first derivative [tex]\( f'(x) \)[/tex]:
[tex]\[
f'(x) = \frac{d}{dx}(x^3 + 9x^2 + 2x - 48) = 3x^2 + 18x + 2
\][/tex]
- Set the first derivative to zero to find critical points:
[tex]\[
3x^2 + 18x + 2 = 0
\][/tex]
- Solve the quadratic equation for [tex]\( x \)[/tex] using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]:
[tex]\[
x = \frac{-18 \pm \sqrt{18^2 - 4 \cdot 3 \cdot 2}}{2 \cdot 3} = \frac{-18 \pm \sqrt{324 - 24}}{6} = \frac{-18 \pm \sqrt{300}}{6} = \frac{-18 \pm 10\sqrt{3}}{6} = \frac{-9 \pm 5\sqrt{3}}{3}
\][/tex]
- Thus, the critical points are:
[tex]\[
x = -3 \pm \frac{5\sqrt{3}}{3}
\][/tex]
3. Evaluate the Function at Critical Points:
- Use these [tex]\( x \)[/tex] values to find the corresponding [tex]\( y \)[/tex] values:
[tex]\[
f\left(-3 + \frac{5\sqrt{3}}{3}\right) \quad \text{and} \quad f\left(-3 - \frac{5\sqrt{3}}{3}\right)
\][/tex]
- Substitute back in to find [tex]\( f(x) \)[/tex] at these points for exact values or numerical approximations.
4. Find Intercepts:
- Y-Intercept: Set [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = -48
\][/tex]
- X-Intercepts: For [tex]\( f(x) = 0 \)[/tex]:
[tex]\[
x^3 + 9x^2 + 2x - 48 = 0
\][/tex]
- Find the roots using methods like factoring, synthetic division, or numerical methods.
5. Analyze End Behavior:
- For [tex]\( x \rightarrow \infty \)[/tex] and [tex]\( x \rightarrow -\infty \)[/tex], cubic functions have:
[tex]\[
\text{As } x \rightarrow \infty, f(x) \rightarrow \infty
\][/tex]
[tex]\[
\text{As } x \rightarrow -\infty, f(x) \rightarrow -\infty
\][/tex]
6. Plot the Graph:
- Use the information from the steps above to sketch the graph.
- Mark the critical points, intercepts, and the general shape considering the end behavior.
By piecing this information together, you should be able to accurately sketch a graph of [tex]\( f(x) = x^3 + 9x^2 + 2x - 48 \)[/tex].
