Question
Complete a sketch of j(x) = 12x^3/5 - 27x/5 and include its end behavior.
Answers
GPT 3.5
To sketch the function j(x) = 12x^(3/5) - 27x/5, we can follow these steps:
Step 1: Determine the end behavior:
To find the end behavior, we look at what happens to the values of j(x) as x approaches positive infinity and negative infinity.
As x approaches positive infinity, the dominant term in j(x) is 12x^(3/5). Since the exponent of x is greater than 1, as x gets larger and larger, the dominant term will also grow without bound. Therefore, the end behavior as x approaches positive infinity is that j(x) approaches positive infinity.
As x approaches negative infinity, the dominant term in j(x) is -27x/5. We can see that as x gets extremely large in the negative direction, the dominant term will also grow without bound but in the negative direction. Therefore, the end behavior as x approaches negative infinity is that j(x) approaches negative infinity.
Step 2: Find the x-intercepts:
To find the x-intercepts, we solve j(x) = 0:
12x^(3/5) - 27x/5 = 0
First, let's factor out a common term:
(12x^(3/5))/5 - (27x/5) = 0
Now, multiply both sides by 5 to eliminate the denominator:
12x^(3/5) - 27x = 0
Factoring out an x:
x(12x^(2/5) - 27) = 0
Setting each factor equal to zero:
x = 0 or 12x^(2/5) - 27 = 0
For the second equation, we can solve for x by isolating the squared term:
12x^(2/5) - 27 = 0
12x^(2/5) = 27
x^(2/5) = 27/12
x^(2/5) = 2.25
Taking the fifth root of both sides:
x = (2.25)^(5/2)
x ≈ ±4.423
Therefore, the x-intercepts are approximately x = 0 and x ≈ ±4.423.
Step 3: Determine the sign of j(x) between the x-intercepts:
To determine the sign of j(x) between the x-intercepts, we can test a value within each interval.
For x < -4.423, we can choose x = -5:
j(-5) = 12(-5)^(3/5) - 27(-5)/5
= -174.33
Since this value is negative, j(x) is negative for x < -4.423.
For -4.423 < x < 0, we can choose x = -1:
j(-1) ≈ 4.370
Since this value is positive, j(x) is positive for -4.423 < x < 0.
For 0 < x < 4.423, we can choose x = 1:
j(1) ≈ -4.370
Since this value is negative, j(x) is negative for 0 < x < 4.423.
For x > 4.423, we can choose x = 5:
j(5) ≈ 174.33
Since this value is positive, j(x) is positive for x > 4.423.
Step 4: Plot the points and sketch the graph:
Now, we can plot the points (x, j(x)) that we found:
(-5, -174.33), (-1, 4.370), (0, 0), (1, -4.370), (5, 174.33)
Based on the sign analysis, between the x-intercepts, the graph of j(x) will start below the x-axis, cross it at x = -1, reach a peak at x = 0, then cross the x-axis again at x = 1, and continue above the x-axis.
The overall shape of the graph will resemble a cubic function with endpoints approaching infinity in opposite directions.
Note: The sketch is just a rough approximation and not to scale.
Here is the rough sketch of j(x) with its end behavior:
```
|
|
|
|
|
----------------|------------------
<-- --> <-- -->
x < -4.423 x > 4.423
```
I hope this helps! Let me know if you have any further questions.
Step 1: Determine the end behavior:
To find the end behavior, we look at what happens to the values of j(x) as x approaches positive infinity and negative infinity.
As x approaches positive infinity, the dominant term in j(x) is 12x^(3/5). Since the exponent of x is greater than 1, as x gets larger and larger, the dominant term will also grow without bound. Therefore, the end behavior as x approaches positive infinity is that j(x) approaches positive infinity.
As x approaches negative infinity, the dominant term in j(x) is -27x/5. We can see that as x gets extremely large in the negative direction, the dominant term will also grow without bound but in the negative direction. Therefore, the end behavior as x approaches negative infinity is that j(x) approaches negative infinity.
Step 2: Find the x-intercepts:
To find the x-intercepts, we solve j(x) = 0:
12x^(3/5) - 27x/5 = 0
First, let's factor out a common term:
(12x^(3/5))/5 - (27x/5) = 0
Now, multiply both sides by 5 to eliminate the denominator:
12x^(3/5) - 27x = 0
Factoring out an x:
x(12x^(2/5) - 27) = 0
Setting each factor equal to zero:
x = 0 or 12x^(2/5) - 27 = 0
For the second equation, we can solve for x by isolating the squared term:
12x^(2/5) - 27 = 0
12x^(2/5) = 27
x^(2/5) = 27/12
x^(2/5) = 2.25
Taking the fifth root of both sides:
x = (2.25)^(5/2)
x ≈ ±4.423
Therefore, the x-intercepts are approximately x = 0 and x ≈ ±4.423.
Step 3: Determine the sign of j(x) between the x-intercepts:
To determine the sign of j(x) between the x-intercepts, we can test a value within each interval.
For x < -4.423, we can choose x = -5:
j(-5) = 12(-5)^(3/5) - 27(-5)/5
= -174.33
Since this value is negative, j(x) is negative for x < -4.423.
For -4.423 < x < 0, we can choose x = -1:
j(-1) ≈ 4.370
Since this value is positive, j(x) is positive for -4.423 < x < 0.
For 0 < x < 4.423, we can choose x = 1:
j(1) ≈ -4.370
Since this value is negative, j(x) is negative for 0 < x < 4.423.
For x > 4.423, we can choose x = 5:
j(5) ≈ 174.33
Since this value is positive, j(x) is positive for x > 4.423.
Step 4: Plot the points and sketch the graph:
Now, we can plot the points (x, j(x)) that we found:
(-5, -174.33), (-1, 4.370), (0, 0), (1, -4.370), (5, 174.33)
Based on the sign analysis, between the x-intercepts, the graph of j(x) will start below the x-axis, cross it at x = -1, reach a peak at x = 0, then cross the x-axis again at x = 1, and continue above the x-axis.
The overall shape of the graph will resemble a cubic function with endpoints approaching infinity in opposite directions.
Note: The sketch is just a rough approximation and not to scale.
Here is the rough sketch of j(x) with its end behavior:
```
|
|
|
|
|
----------------|------------------
<-- --> <-- -->
x < -4.423 x > 4.423
```
I hope this helps! Let me know if you have any further questions.