Question

What is the rule of the cubic function for which the graph passes through the points with coordinates (0,135), (1,156), (2,115), (3,0)?

Answer 1

See below.

Explanation:

The simpler way to solve this question, assuming the cubic function is given by

`p(x) = a x^3+b x^2+c x+d` and having the table

`(x_i,y_i), i = 1,2,3,4` is by solving the system of linear equations

`{(a x_1^3+b x_1^2+c x_1+d = y_1),(a x_2^3+b x_2^2+c x_2+d = y_2),(a x_3^3+b x_3^2+c x_3+d = y_3),(a x_4^3+b x_4^2+c x_4+d = y_4):}`

or

`((x_1^3,x_1^2,x_1,1),(x_2^3,x_2^2,x_2,1),(x_3^3,x_3^2,x_3,1),(x_4^3,x_4^2,x_4,1))((a),(b),(c),(d)) = ((y_1),(y_2),(y_3),(y_4))`

Solving for `a,b,c,d` we obtain the solution.

`p(x)=-2x^3-25x^2+48x+135`

Attached a plot showing the interpolation.

Answer 2

`f(x) = -2x^3-25x^2+48x+135`

Explanation:

Since the `x` coordinates are sequential integers, we can use a difference method to find the formula:

Write down the initial sequence of `y` coordinates:

`color(blue)(135), 156, 115, 0`

Write down the sequence of differences between successive terms:

`color(blue)(21), -41, -115`

Write down the sequence of differences of those differences:

`color(blue)(-62), -74`

Write down the sequence of differences of those differences:

`color(blue)(-12)`

Having reached a constant sequence (albeit of just one term), we can use the initial term of each of these sequences as coefficients to find the formula:

`f(x) = color(blue)(135)/(0!)+color(blue)(21)/(1!)x+color(blue)(-62)/(2!)x(x-1)+color(blue)(-12)/(3!)x(x-1)(x-2)`

`color(white)(f(x)) = 135+21x-31x^2+31x-2x^3+6x^2-4x`

`color(white)(f(x)) = -2x^3-25x^2+48x+135`