Find the equation of the curve that joins the following points. Hint: Use common differences.

a. The zeros of the function are 3,-1,-5. The y-intercept is 15. Find the function.

b. (1,1), (0,1), (2,-5), (-1,-5)

a. x = 3

x-3 = 0

x = -1
x+1 = 0

x = -5
x+5 = 0

Y = (x-3)(x+1)(x+5) = 0
Multiply the 1st two parenthesis:
Y = (x^2-2x-3)(x+5) = 0
Eq: Y = x^3 + 3x^2 -13x - 15 = 0

a. To find the equation of the curve that joins the given points, we can start by determining the common differences.

The common difference between consecutive zeros is 4 (3 - (-1) = 4) and between the remaining zeros is also 4 (-1 - (-5) = 4).

Since the y-intercept is given as 15, we know the function intersects the y-axis at (0, 15).

Now, we can use the common differences and the y-intercept to write the equation in factored form.

The factored form of the equation is:

f(x) = a(x - 3)(x + 1)(x + 5)

To find the value of "a" in the equation, we substitute the coordinates of the y-intercept (0, 15) into the equation:

15 = a(0 - 3)(0 + 1)(0 + 5)
15 = -15a

Solving for "a":

-15a = 15
a = -1

Thus, the equation of the curve is:

f(x) = -1(x - 3)(x + 1)(x + 5)

b. To find the equation of the curve that joins the given points, we can use the method of interpolation.

First, we can determine the common differences in the x- and y-coordinates.

x-coordinates: 1 - 0 = 1, 2 - 0 = 2, -1 - 0 = -1
y-coordinates: 1 - 1 = 0, -5 - 1 = -6, -5 - 1 = -6

Since the y-coordinate difference is constant at -6, we can deduce that the equation of the curve will be of the form y = ax^2 + bx + c.

To find the values of a, b, and c, we can substitute the coordinates of any of the given points into the equation.

For example, using the point (1, 1):

1 = a(1^2) + b(1) + c

Simplifying the equation:

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

Similarly, substituting the coordinates of another point, say (0, 1):

1 = a(0^2) + b(0) + c

Simplifying the equation:

1 = c -------------- Equation (2)

Substituting the value of c = 1 into Equation (1):

1 = a + b + 1
a + b = 0 -------------- Equation (3)

Now, we have two equations with two variables, a and b.

Equation (2) tells us c = 1, which means the equation of the curve is y = ax^2 + bx + 1.

To find the values of a and b, we can substitute Equation (3) into Equation (2):

b = -a

Substituting this relationship into Equation (3):

-a + (-a) = 0
-2a = 0

Solving for "a":
a = 0

Since a = 0, the equation of the curve becomes:

y = 0x^2 + bx + 1
y = bx + 1

To find the value of "b", we can substitute the coordinates of any of the given points. Let's use (2, -5):

-5 = b(2) + 1
-5 = 2b + 1

Solving for "b":
2b = -6
b = -3

Thus, the equation of the curve is:
y = -3x + 1