To sketch the graph of the function 𝑓(𝑥)=𝑥³−6⋅𝑥², we need to find the following information:

1. Domain and Range
2. Intercepts
3. Critical points
4. Inflection points
5. Asymptotes
6. End behavior

Let's find these step by step:

1. Domain and Range:
The domain of a polynomial function is all real numbers, so the domain of this function is (-∞, ∞). The range of a cubic function is also all real numbers, so the range is also (-∞, ∞).

2. Intercepts:
To find the x-intercepts, set 𝑓(𝑥) = 0 and solve for x.
𝑥³−6⋅𝑥² = 0
𝑥²(𝑥−6) = 0
So, x = 0 and x = 6 are the x-intercepts.
The y-intercept is found by setting x = 0 in the function, so y = 0.

3. Critical points:
These are found by taking the derivative of the function and setting it equal to zero.
𝑓'(𝑥) = 3𝑥² - 12𝑥 = 0
3𝑥(𝑥 - 4) = 0
So, x = 0 and x = 4 are the critical points.

4. Inflection points:
These are found by taking the second derivative of the function and setting it equal to zero.
𝑓''(𝑥) = 6𝑥 - 12 = 0
So, x = 2 is the inflection point.

5. Asymptotes:
Since this is a polynomial function, it does not have any vertical or horizontal asymptotes.

6. End behavior:
Since the leading term is 𝑥³ and the coefficient is positive, as x approaches -∞, 𝑓(𝑥) approaches -∞ and as x approaches ∞, 𝑓(𝑥) approaches ∞.

Now, you can use this information to sketch the graph or select the correct graph from a list.

Question

To sketch the graph of the function 𝑓(𝑥)=𝑥³−6⋅𝑥², we need to find the following information:

1. Domain and Range
2. Intercepts
3. Critical points
4. Inflection points
5. Asymptotes
6. End behavior

Let's find these step by step:

1. Domain and Range: 
   The domain of a polynomial function is all real numbers, so the domain of this function is (-∞, ∞). The range of a cubic function is also all real numbers, so the range is also (-∞, ∞).

2. Intercepts: 
   To find the x-intercepts, set 𝑓(𝑥) = 0 and solve for x. 
   𝑥³−6⋅𝑥² = 0
   𝑥²(𝑥−6) = 0
   So, x = 0 and x = 6 are the x-intercepts.
   The y-intercept is found by setting x = 0 in the function, so y = 0.

3. Critical points: 
   These are found by taking the derivative of the function and setting it equal to zero.
   𝑓'(𝑥) = 3𝑥² - 12𝑥 = 0
   3𝑥(𝑥 - 4) = 0
   So, x = 0 and x = 4 are the critical points.

4. Inflection points: 
   These are found by taking the second derivative of the function and setting it equal to zero.
   𝑓''(𝑥) = 6𝑥 - 12 = 0
   So, x = 2 is the inflection point.

5. Asymptotes: 
   Since this is a polynomial function, it does not have any vertical or horizontal asymptotes.

6. End behavior: 
   Since the leading term is 𝑥³ and the coefficient is positive, as x approaches -∞, 𝑓(𝑥) approaches -∞ and as x approaches ∞, 𝑓(𝑥) approaches ∞.

Now, you can use this information to sketch the graph or select the correct graph from a list.

Knowee AI · Accepted Answer

To sketch the graph of the function 𝑓(𝑥)=𝑥³−6⋅𝑥², we need to find the following information:

1. Domain and Range
2. Intercepts
3. Critical points
4. Inflection points
5. Asymptotes
6. End behavior

Let's find these step by step:

1. Domain and Range: 
   The domain of a polynomial function is all real numbers, so the domain of this function is (-∞, ∞). The range of a cubic function is also all real numbers, so the range is also (-∞, ∞).

2. Intercepts: 
   To find the x-intercepts, set 𝑓(𝑥) = 0 and solve for x. 
   𝑥³−6⋅𝑥² = 0
   𝑥²(𝑥−6) = 0
   So, x = 0 and x = 6 are the x-intercepts.
   The y-intercept is found by setting x = 0 in the function, so y = 0.

3. Critical points: 
   These are found by taking the derivative of the function and setting it equal to zero.
   𝑓'(𝑥) = 3𝑥² - 12𝑥 = 0
   3𝑥(𝑥 - 4) = 0
   So, x = 0 and x = 4 are the critical points.

4. Inflection points: 
   These are found by taking the second derivative of the function and setting it equal to zero.
   𝑓''(𝑥) = 6𝑥 - 12 = 0
   So, x = 2 is the inflection point.

5. Asymptotes: 
   Since this is a polynomial function, it does not have any vertical or horizontal asymptotes.

6. End behavior: 
   Since the leading term is 𝑥³ and the coefficient is positive, as x approaches -∞, 𝑓(𝑥) approaches -∞ and as x approaches ∞, 𝑓(𝑥) approaches ∞.

Now, you can use this information to sketch the graph or select the correct graph from a list.

Consider the function 𝑓(𝑥)=𝑥3−6⋅𝑥2. Find the relevant curve sketching information and then select the correct graph from the list below.

Question

Solution

Similar Questions

Upgrade your grade with Knowee