Writing Cubic Functions Cubic Function A cubic function

Writing Cubic Functions

Cubic Function • A cubic function is a function of the form f(x) = ax 3 + bx 2 + cx + d • A linear function contains a polynomial with degree one (mx 1 + b) and a quadratic function contains a polynomial with degree two (ax 2 + bx + c). • A cubic function contains a polynomial with degree three (ax 3 + bx 2 + cx + d).

Cubic Functions • With linear functions, we looked at first finite differences. • With quadratic functions, we looked at second finite differences • With cubic functions, what will we look for? • THIRD FINITE DIFFERENCES

Cubic Functions • Just like with quadratic functions, with cubic functions we will find our necessary values (a, b, c, d) by looking at the finite differences • The value of d is the y-intercept (when x = 0) • The third difference is equal to 6 a • The second difference between the FIRST TWO pairs of y-values is equal to 6 a + 2 b • The first difference between the y-values for x = 0 and x = 1 is equal to a + b + c

Examples • Use the table to determine the cubic function rule for the table.

Examples • Step 1: Analyze the first, second, and third finite differences to calculate the values of a, b, c, and d.

Examples • Step 2: The value of d is the y value when x = 0; d = 0 • Step 3: The third difference is equal to 6 a. Since 6 a = 36, a = 6 • Step 4: The second difference between the first two pairs of values (x = 0 and x = 1; x = 1 and x = 2) is 6 a + 2 b • 6 a + 2 b = 36 • 36 + 2 b = 36 • 2 b = 0 • b=0 y-

Examples • Step 5: The first difference between the y-values for x = 0 and x = 1 is equal to a + b + c • a+b+c=6 • 6+0+c=6 • c=0 • Step 6: Write the cubic function rule • y = 6 x 3 + 0 x 2 + 0 x + 0 or y = 6 x 3

Examples • Write a cubic function for the values in the table.

Examples • Step 1: Analyze the first, second, and third finite differences to calculate the values of a, b, c, and d.

Examples •