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Find the equation of a cubic function that passes through the points (-4,2) and (-3,0) and is tangent to the x-axis at the point (a,0).

User Ppetrov
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Final answer:

To find the equation of the cubic function, we need to solve a system of equations using the points (-4,2) and (-3,0), and the fact that the function is tangent to the x-axis at (a,0). Therefore, the cubic equation is -9a^3 - 18a^2 + a + 54 = 0.

Step-by-step explanation:

To find the equation of a cubic function that passes through the points (-4,2) and (-3,0), we can use the general form of a cubic function: y = ax^3 + bx^2 + cx + d. Plugging in the coordinates of the two points, we get the following system of equations:

-4a + 16b - 48c + d = 2 -3a + 9b - 27c + d = 0

To find the value of a, we can use the fact that the cubic function is tangent to the x-axis at the point (a,0). This means that the slope of the function at x = a is zero.

Taking the derivative of the cubic function and setting it equal to zero, we get:

3ax^2 + 2bx + c = 0

Plugging in x = a, we get:

3a^2 + 2ba + c = 0

We can now solve the system of equations to find a, b, c, and d. Solving for d first, we get:

d = 27c - 9b + 3a

Substituting this into the first equation, we get:

-4a + 16b - 48c + 27c - 9b + 3a = 2

Simplifying, we get:

-a + 7b - 21c = -2/3

Solving for b in terms of a and c, we get:

b = (2a - 42c)/7

Substituting this into the second equation, we get:

-3a + 9(2a - 42c)/7 - 27c + 27c - 9(2a - 42c)/7 + 3a = 0

Simplifying, we get:

-6a + 54c = 0

Solving for a in terms of c, we get:

a = -9c

Substituting this into the equation for b, we get:

b = -18c

Substituting these values into the equation for d, we get:

d = 54c

Therefore, the equation of the cubic function is:

y = -9cx^3 - 18cx^2 + cx + 54c

To find the value of a, we can use the fact that the cubic function is tangent to the x-axis at the point (a,0). This means that the slope of the function at x = a is zero. Taking the derivative of the cubic function and setting it equal to zero, we get:

-27ax^2 - 36bx - 9c = 0

Plugging in x = a, we get:

-27a^2 - 36ba - 9c = 0

Since the cubic function is tangent to the x-axis at the point (a,0), we know that the y-coordinate of this point is zero. Plugging in x = a and y = 0 into the equation of the cubic function, we get:

0 = -9ca^3 - 18ca^2 + ca + 54c

Simplifying, we get:

-9a^3 - 18a^2 + a + 54 = 0

Therefore, the cubic equation is -9a^3 - 18a^2 + a + 54 = 0.

User Matt Whitlock
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