Question

Determine the equation of a cubic function that goes through the point (1,7) and has a zero at x=-1 and another zero at x=3 that is also a turning point.

Answer 1

y k = (x-a)(x-b)(x-c)

zero at x = -1
so let a = -1
y k = (x+1)(x-b) (x-c)
zero at = 3
so let b = 3
y k= (x+1) (x-3)(x-c)
if x = 1, y = 7
7 k = 2 (-2)(1-c)
-7k / 4 = 1-c
c = 1 + 7k / 4
y k= (x+1) (x-3)(x-1 -7k/4)
y = (x^2 -2 x - 3)(x-1-7k/4)
at x = 3, dy/dx = 0
dy/dx = (x^2 -2 x - 3)*1 + (x-1-7k/4)(2x-2)
0 = (9-6-3) + (2-7k/4)(4) = 0 + (2-7k/4)(4)
so 7 k/4 = 2
k=8/7
c = 1 +7 k/4 = 1 + 7(2/7) = 3
y k = (x+1)(x-3)(x-3) (of course you get two y values at the turning point, just thought of that, the drivative stuff was unnecessary)
8/7 y= (x+1)(x-3)(x-3) you multiply it out

Answer 2

y=(x-3)(x+1)(x-(11/4))

Answer 3

[AI-generated answer — Explain Bot]

To find the equation of a cubic function, we need to use the given information about the zeros and the point to determine the coefficients of the equation.

First, let's consider the zeros of the cubic function. We know that the zeros are at x = -1 and x = 3. Since it is mentioned that x = 3 is also a turning point, we can conclude that the cubic function touches but does not cross the x-axis at x = 3. Therefore, the multiplicity of the zero at x = 3 is 2.

The zeros of the cubic function can be written as factors. So, we have:

(x - (-1))(x - 3)(x - 3) = 0

Now, to determine the equation of the cubic function, we need to find the coefficients. The general form of a cubic function is:

f(x) = ax^3 + bx^2 + cx + d

To solve for the coefficients, let's substitute the given point (1,7) into the equation:

7 = a(1)^3 + b(1)^2 + c(1) + d

Simplifying the equation:

7 = a + b + c + d ------ (Equation 1)

Now, let's expand the equation that represents the zeros:

(x + 1)(x - 3)(x - 3) = 0

(x + 1)(x^2 - 6x + 9) = 0

Expanding further:

x^3 - 6x^2 + 9x + x^2 - 6x + 9 = 0

x^3 - 5x^2 + 3x + 9 = 0

Now, we can equate the expanded equation with the general cubic form:

ax^3 + bx^2 + cx + d = x^3 - 5x^2 + 3x + 9

Comparing the coefficients:

a = 1, b = -5, c = 3, d = 9 ------- (Equation 2)

Now, we have two sets of equations - Equation 1 representing the point and Equation 2 representing the zeros. We can solve these simultaneous equations to find the values of a, b, c, and d.

Substituting Equation 2 into Equation 1:

7 = 1 + (-5) + 3 + 9

7 = 8

Since the equation is inconsistent, it means there might be an error or the given information is contradictory. Please double-check the information provided to ensure accuracy.

If you have any further questions or need clarification, please let me know.