Answer and Step-by-step explanation:
To find "b" and write the equation in standard form, we need to use the given points and the general form of the function:
h(x) = (1/b(x - h))³ + k
First, we can use the point (-5, -5):
-5 = (1/b((-5) - h))³ + k
Next, we can use the point (1, -2):
-2 = (1/b((1) - h))³ + k
Finally, we can use the point (-2, -3):
-3 = (1/b((-2) - h))³ + k
We can use these three equations to solve for "b" and "h". One way to do this is to subtract the third equation from the first equation, which gives:
2 = (1/b(3 - h))³
Taking the cube root of both sides and multiplying by b gives:
b³ = 1/8
b = 1/2
Now that we have found "b", we can use the second equation to solve for "h":
-2 = (1/2(1 - h))³ + k
Simplifying this equation and using the fact that k is a constant, we get:
-8 = (1 - h)³ + 8k
-1 = (1 - h)³ + k
We can use this equation along with the point (-5, -5) to solve for "h":
-5 = (1/2((-5) - h))³ + k
-5 = (1/2(-5 - h))³ + k
-5 = (1/2(-5 - h))³ + (-1/8)
-40 = (-5 - h)³
Taking the cube root of both sides and multiplying by -1 gives:
h = 5 - ∛(-40)
h = 5 + 2∛10
Now we have found "b" and "h", and we can use them to write the equation in standard form. First, we substitute the value of "b" into the general form of the function:
h(x) = (1/(1/2(x - (5 + 2∛10))))³ + k
Simplifying this expression and using the fact that k is a constant, we get:
h(x) = 8(x - (5 + 2∛10))³ + k
This is the equation in standard form. We could expand and simplify the expression further, but this is the final answer.