Complete a sketch of j(x) = 12x^3/5 - 27x/5 and include its end behavior.
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:
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<-- --> <-- -->
x < -4.423 x > 4.423
```
I hope this helps! Let me know if you have any further questions.
To complete a sketch of the given function j(x) = 12x^(3/5) - 27x/5, we can follow these steps:
Step 1: Find the y-intercept:
To find the y-intercept, substitute x = 0 into the equation:
j(0) = 12(0)^(3/5) - 27(0)/5 = 0
So, the y-intercept is (0, 0).
Step 2: Find the x-intercept(s):
To find the x-intercept(s), set j(x) equal to zero and solve for x:
12x^(3/5) - 27x/5 = 0
Multiply through by 5x to eliminate the denominator:
12x^(3/5)(5x) - 27x = 0
60x^(4/5) - 27x = 0
Factor out x:
x(60x^(4/5) - 27) = 0
Setting each factor equal to zero, we have two possibilities:
x = 0, or 60x^(4/5) - 27 = 0
Solving for x in the second equation:
60x^(4/5) = 27
Divide through by 60:
x^(4/5) = 27/60
x^(4/5) = 9/20
Raise both sides to the power of 5/4:
(x^(4/5))^(5/4) = (9/20)^(5/4)
x = (9/20)^(5/4)
Using a calculator, we find that x ≈ 0.6164.
So, the x-intercept is approximately (0.6164, 0).
Step 3: Find critical points:
To find the critical points, we need to find the values of x where the derivative of j(x) is equal to zero or undefined:
The derivative of j(x) is given by:
j'(x) = (36/5)x^(-2/5) - 27/5
Setting j'(x) equal to zero:
(36/5)x^(-2/5) - 27/5 = 0
Multiply through by 5 to eliminate the fractions:
36x^(-2/5) - 27 = 0
Simplify further:
36x^(-2/5) = 27
Divide through by 36:
x^(-2/5) = 3/4
Take the reciprocal of both sides:
x^(2/5) = 4/3
Raise both sides to the power of 5/2:
(x^(2/5))^(5/2) = (4/3)^(5/2)
x^1 = (4/3)^(5/2)
x = (4/3)^(5/2)
Using a calculator, we find that x ≈ 0.8536.
So, the critical point is approximately (0.8536, j(0.8536)).
Step 4: Determine the end behavior:
The end behavior of a function describes its behavior as x approaches positive infinity and negative infinity.
As x approaches positive infinity, the highest power of x dominates the expression. In this case, it is x^(3/5). Since the exponent is positive, as x approaches positive infinity, j(x) will also approach positive infinity.
As x approaches negative infinity, we need to consider the parity of the exponent on the x-term. In this case, the highest power of x is x^(3/5), which is odd. Therefore, as x approaches negative infinity, j(x) will also approach negative infinity.
Step 5: Plot the points on a graph and sketch the curve:
Using the information obtained from the steps above, we can plot the points (0,0), (0.6164, 0), and (0.8536, j(0.8536)) on a graph and sketch the curve accordingly. The end behavior indicates that the curve will rise on the right side and fall on the left side.
Note: It may be helpful to use graphing software or a graphing calculator to visualize the sketch accurately.
To complete a sketch of the function j(x) = 12x^(3/5) - 27x/5, we can follow these steps:
1. Start by identifying the key features of the function:
- The exponent of x is 3/5, indicating that the function has a fractional exponent, making it a root function.
- The function is a polynomial consisting of two terms.
2. Determine the x-intercepts:
- To find the x-intercepts, set j(x) = 0 and solve for x.
- 12x^(3/5) - 27x/5 = 0
- Factor out common terms: x(12x^(3/5) - 27/5) = 0
- From the equation, we find that x = 0 and 12x^(3/5) - 27/5 = 0.
3. Determine the y-intercept:
- To find the y-intercept, plug in x = 0 into the equation: j(0) = 12(0)^(3/5) - 27(0)/5 = 0.
4. Analyze the end behavior:
- The end behavior refers to what happens to the graph as x approaches positive or negative infinity.
- Since the leading term of the polynomial is 12x^(3/5), and the exponent is positive, the end behavior will be as follows:
- As x approaches positive infinity, j(x) also approaches positive infinity.
- As x approaches negative infinity, j(x) also approaches negative infinity.
5. Sketch the graph:
- Plot the x-intercepts, y-intercept, and any other points of interest on the graph.
- Considering the end behavior, draw a smooth curve that passes through these points, reflecting the shape of the function.
Keep in mind that without a specific range or restrictions for x, the sketch is a general representation of the given function.