It looks like the ODE is
with the initial condition of .
Rewrite the right side in terms of the unit step function,
In this case, we have
The Laplace transform of the step function is easy to compute:
So, taking the Laplace transform of both sides of the ODE, we get
Solve for :
We can split the first term into partial fractions:
If , then .
If , then .
Take the inverse transform of both sides, recalling that
where is the Laplace transform of the function . We have
We then end up with
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Answer:
where a>0.
To graph the the polynomial, begin in the left top of quadrant 2. Then draw downwards to the first real zero on the x-axis at -2. Cross the x-axis and then curve back up to 1/2 on the x-axis. Cross through again and curve back down to cross for the last time at 3 on the x-axis. The graph then ends going down towards the right in quadrant 4. It forms an s shape.
Step-by-step explanation:
The real zeros are the result of setting each factor of the polynomial to zero. By reversing this process, we find:
- zero 1/2 is factor (2x-1)
We write them together with an unknown leading coefficient a which is negative so -a.
where a>0
The leading coefficient of a polynomial determines the direction of the graph's end behavior.
- A positive leading coefficient has the end behavior point up when an even degree and point opposite directions when an odd degree with the left down and the right up.
- A negative leading coefficient has the end behavior point down when an even degree and point opposite directions when an odd degree with the left up and the right down.
- This graph has all odd multiplicity. The graph will cross through the x-axis each time at its real zeros.
To graph the the polynomial, begin in the left top of quadrant 2. Then draw downwards to the first real zero on the x-axis at -2. Cross the x-axis and then curve back up to 1/2 on the x-axis. Cross through again and curve back down to cross for the last time at 3 on the x-axis. The graph then ends going down towards the right in quadrant 4. It forms an s shape.
Given:
Translation of x represented by the translation rule <-6,8>.
To find:
The correct statement for the given rule of translation.
Solution:
Rule of translation is <-6,8>.
Here, x-coordinate represents horizontal shift and y-coordinate represents vertical shift.
x-coordinate is -6, which is negative. So, the figure translated 6 units left.
y-coordinate is 8, which is positive. So, the figure translated 8 units up.
Thus, translation in words is defined as "6 units to the left and 8 units up".
Therefore, the correct option is B.