Answer :
To graph the function [tex]\( f(x) = x^3 - 2x^2 - x + 2 \)[/tex], follow these detailed steps:
### 1. Polynomial Analysis:
The given function is a cubic polynomial of the form [tex]\( f(x) = x^3 + bx^2 + cx + d \)[/tex]. Here:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -2 \)[/tex]
- [tex]\( c = -1 \)[/tex]
- [tex]\( d = 2 \)[/tex]
### 2. Identifying Key Features:
#### End Behavior:
For a cubic polynomial of the form [tex]\( x^3 \)[/tex], its end behavior is:
- As [tex]\( x \to \infty \)[/tex], [tex]\( f(x) \to \infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to -\infty \)[/tex].
#### Derivative and Critical Points:
To find the critical points, calculate the first derivative:
[tex]\[ f'(x) = 3x^2 - 4x - 1 \][/tex]
Set the first derivative to zero to find the critical points:
[tex]\[ 3x^2 - 4x - 1 = 0 \][/tex]
This is a quadratic equation. Solve for [tex]\( x \)[/tex] using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]:
[tex]\[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(3)(-1)}}{2(3)} \][/tex]
[tex]\[ x = \frac{4 \pm \sqrt{16 + 12}}{6} \][/tex]
[tex]\[ x = \frac{4 \pm \sqrt{28}}{6} \][/tex]
[tex]\[ x = \frac{4 \pm 2\sqrt{7}}{6} \][/tex]
[tex]\[ x = \frac{2 \pm \sqrt{7}}{3} \][/tex]
Thus, the critical points are [tex]\( x = \frac{2 + \sqrt{7}}{3} \)[/tex] and [tex]\( x = \frac{2 - \sqrt{7}}{3} \)[/tex].
#### Second Derivative and Concavity:
To determine the concavity, calculate the second derivative:
[tex]\[ f''(x) = 6x - 4 \][/tex]
Set the second derivative to zero to find possible inflection points:
[tex]\[ 6x - 4 = 0 \][/tex]
[tex]\[ 6x = 4 \][/tex]
[tex]\[ x = \frac{2}{3} \][/tex]
Evaluating the concavity around [tex]\( x = \frac{2}{3} \)[/tex]:
- For [tex]\( x < \frac{2}{3} \)[/tex], [tex]\( f''(x) < 0 \)[/tex], the function is concave down.
- For [tex]\( x > \frac{2}{3} \)[/tex], [tex]\( f''(x) > 0 \)[/tex], the function is concave up.
### 3. Important Points:
Calculate the function's value at the critical points and at [tex]\( x = \frac{2}{3} \)[/tex]:
[tex]\[ f\left(\frac{2 + \sqrt{7}}{3}\right) \][/tex]
[tex]\[ f\left(\frac{2 - \sqrt{7}}{3}\right) \][/tex]
[tex]\[ f\left(\frac{2}{3}\right) = \left(\frac{2}{3}\right)^3 - 2\left(\frac{2}{3}\right)^2 - \left(\frac{2}{3}\right) + 2 = \frac{8}{27} - \frac{8}{9} - \frac{2}{3} + 2 \][/tex]
### 4. Plotting the Function:
Use the above key points, end behavior, and the shape of cubic polynomials to sketch the graph:
- Plot points such as [tex]\( f(x) \)[/tex] at critical values.
- Note the locations where the function crosses the x-axis, if applicable.
- Sketch the end behavior.
### 5. Example Shape:
The graph will show the typical cubic behavior, going from negative to positive infinity. It might have one or two turning points depending on the nature of the critical points found. The concavity will change around [tex]\( x = \frac{2}{3} \)[/tex]. Therefore, the function is expected to rise steeply as [tex]\( x \)[/tex] increases and decrease steeply as [tex]\( x \)[/tex] decreases.
### Final Sketch:
The graph of [tex]\( f(x) = x^3 - 2x^2 - x + 2 \)[/tex] will have the following characteristics:
- Starts from [tex]\( -\infty \)[/tex] in the left-hand side
- Rises, turns around [tex]\( x = \frac{2 - \sqrt{7}}{3} \)[/tex], becomes concave down up to [tex]\( x = \frac{2}{3} \)[/tex], turns around [tex]\( x = \frac{2 + \sqrt{7}}{3} \)[/tex]
- Changes concavity at [tex]\( x = \frac{2}{3} \)[/tex], finally rising to [tex]\( \infty \)[/tex].
This general shape emphasizes the cubic nature with shifts in concavity and critical points.
### 1. Polynomial Analysis:
The given function is a cubic polynomial of the form [tex]\( f(x) = x^3 + bx^2 + cx + d \)[/tex]. Here:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -2 \)[/tex]
- [tex]\( c = -1 \)[/tex]
- [tex]\( d = 2 \)[/tex]
### 2. Identifying Key Features:
#### End Behavior:
For a cubic polynomial of the form [tex]\( x^3 \)[/tex], its end behavior is:
- As [tex]\( x \to \infty \)[/tex], [tex]\( f(x) \to \infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to -\infty \)[/tex].
#### Derivative and Critical Points:
To find the critical points, calculate the first derivative:
[tex]\[ f'(x) = 3x^2 - 4x - 1 \][/tex]
Set the first derivative to zero to find the critical points:
[tex]\[ 3x^2 - 4x - 1 = 0 \][/tex]
This is a quadratic equation. Solve for [tex]\( x \)[/tex] using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]:
[tex]\[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(3)(-1)}}{2(3)} \][/tex]
[tex]\[ x = \frac{4 \pm \sqrt{16 + 12}}{6} \][/tex]
[tex]\[ x = \frac{4 \pm \sqrt{28}}{6} \][/tex]
[tex]\[ x = \frac{4 \pm 2\sqrt{7}}{6} \][/tex]
[tex]\[ x = \frac{2 \pm \sqrt{7}}{3} \][/tex]
Thus, the critical points are [tex]\( x = \frac{2 + \sqrt{7}}{3} \)[/tex] and [tex]\( x = \frac{2 - \sqrt{7}}{3} \)[/tex].
#### Second Derivative and Concavity:
To determine the concavity, calculate the second derivative:
[tex]\[ f''(x) = 6x - 4 \][/tex]
Set the second derivative to zero to find possible inflection points:
[tex]\[ 6x - 4 = 0 \][/tex]
[tex]\[ 6x = 4 \][/tex]
[tex]\[ x = \frac{2}{3} \][/tex]
Evaluating the concavity around [tex]\( x = \frac{2}{3} \)[/tex]:
- For [tex]\( x < \frac{2}{3} \)[/tex], [tex]\( f''(x) < 0 \)[/tex], the function is concave down.
- For [tex]\( x > \frac{2}{3} \)[/tex], [tex]\( f''(x) > 0 \)[/tex], the function is concave up.
### 3. Important Points:
Calculate the function's value at the critical points and at [tex]\( x = \frac{2}{3} \)[/tex]:
[tex]\[ f\left(\frac{2 + \sqrt{7}}{3}\right) \][/tex]
[tex]\[ f\left(\frac{2 - \sqrt{7}}{3}\right) \][/tex]
[tex]\[ f\left(\frac{2}{3}\right) = \left(\frac{2}{3}\right)^3 - 2\left(\frac{2}{3}\right)^2 - \left(\frac{2}{3}\right) + 2 = \frac{8}{27} - \frac{8}{9} - \frac{2}{3} + 2 \][/tex]
### 4. Plotting the Function:
Use the above key points, end behavior, and the shape of cubic polynomials to sketch the graph:
- Plot points such as [tex]\( f(x) \)[/tex] at critical values.
- Note the locations where the function crosses the x-axis, if applicable.
- Sketch the end behavior.
### 5. Example Shape:
The graph will show the typical cubic behavior, going from negative to positive infinity. It might have one or two turning points depending on the nature of the critical points found. The concavity will change around [tex]\( x = \frac{2}{3} \)[/tex]. Therefore, the function is expected to rise steeply as [tex]\( x \)[/tex] increases and decrease steeply as [tex]\( x \)[/tex] decreases.
### Final Sketch:
The graph of [tex]\( f(x) = x^3 - 2x^2 - x + 2 \)[/tex] will have the following characteristics:
- Starts from [tex]\( -\infty \)[/tex] in the left-hand side
- Rises, turns around [tex]\( x = \frac{2 - \sqrt{7}}{3} \)[/tex], becomes concave down up to [tex]\( x = \frac{2}{3} \)[/tex], turns around [tex]\( x = \frac{2 + \sqrt{7}}{3} \)[/tex]
- Changes concavity at [tex]\( x = \frac{2}{3} \)[/tex], finally rising to [tex]\( \infty \)[/tex].
This general shape emphasizes the cubic nature with shifts in concavity and critical points.
