Kristen Robinson

EMAT 4680

9 November 2000

Assignment 18 Number 2

The standard form of the function y = a(x^3) + b(x^2) +cx was chosen for observation with respect to the values of a, b, and c. There are four possible basic forms conceivable given the standard form of the function. A constant function, a linear function, a quadratic function, and a cubic function all can result depending upon the values of a, b, and c. The only condition in which a constant function can result is when a = 0, b = 0, and c = 0. Then y = 0 which is a constant function lying on the x-axis.

When a = 0, b = 0, and c does not equal 0, y = cx which is a linear function. This function has a y-intercept at 0. There is a single x-intercept at 0. The slope of the line is equal to c. If c is a positive number, the function is increasing. If c is a negative number, the function is decreasing. The first derivative of the function is y’ = c, a constant function with no x-intercept and a y-intercept at y = c. The second derivative of the function is y" = 0.

When a = 0, c = 0, and b does not equal 0, y = bx^2 which is a quadratic function. This function has a vertex at (0,0). The first derivative of the function is y’ = 2bx, a linear function with x- and y-intercepts at 0 and a slope of 2b. The second derivative of the function is y" = 2b, a constant function with no x-intercept and a y-intercept at y = 2b. If b is a positive number, the original function is concave up. If b is a negative number, the original function is concave down.

When b = 0, c = 0, and a does not equal 0, y = ax^3 which is a cubic function. This function has a point of inflection at (0,0). The first derivative of the function is y’ = 3ax^2 which is a quadratic function with a vertex at (0,0). The second derivative of the function is y" = 6ax which is a linear function with x- and y-intercepts at 0 and a slope of 6a. If a is a positive number, the original function is concave up when x > 0 and concave down when x < 0. If a is a negative number, the original function is concave down when x > 0 and concave up when x < 0.

If a = 0 and both b and c are not equal to 0, y = bx^2 + cx which is a quadratic function. The function has a y-intercept at 0. The function has x-intercepts at 0 and -c/b. The vertex of the function is at (-c/2b, (-c/2) - (c^2/2b)). The first derivative of the function is y’ = 2bx + c which is a linear function with y-intercept at c and a slope of 2b. The second derivative of the function is y" = 2b which is a constant function with no x-intercept and a y-intercept at 2b. If b is positive, the original function is concave up. If b is negative, the original function is concave down.

If b = 0 and both a and c are not equal to 0, y = ax^3 + cx which is a cubic function. The function has a y-intercept at 0. The function has x-intercepts at 0 and at (+/-)sqrt(-c/a). The point of inflection of the function is at (0,0). The first derivative of the function is y’ = 3ax^2 + c which is a quadratic function with vertex (0,c). The second derivative of the function is y" = 6ax which is a linear function with a y-intercept at 0, an x-intercept at 0, and a slope of 6a.

If c = 0 and both a and b are not equal to 0, y = ax^3 + bx^2 which is a cubic function. The function has a y-intercept at 0. The function has x-intercepts at 0 and at -b/a. The point of inflection of the function is at (-b/3a, 2b^3/27a^2). The first derivative of the function is y’ = 3ax^2 + 2bx which is a quadratic function with vertex (-b/3a, -b^2/3a). The second derivative of the function is y" = 6ax + 2b which is a linear function with y-intercept at 2b, x-intercept at -b/3a, and a slope of 6a.

If a is a positive number, the function is a cubic function regardless of whether the values of b and c are positive or negative. The function is concave up when x > -b/3a. The function is concave down when x < -b/3a. If a is a negative number the function is a cubic function regardless of whether the values of b and c are positive or negative. The function is concave up when x < -b/3a. The function is concave down when x > -b/3a.