Here are the steps to graph a cubic function. Thus, the cubic function f(x) = ax 3 + bx 2 + cx + d has inflection point at (-b/3a, f(-b/3a)). To find the critical points of a cubic function f(x) = ax 3 + bx 2 + cx + d, we set the second derivative to zero and solve. The inflection points of a function are the points where the function changes from either "concave up to concave down" or "concave down to concave up".
f(x) has no critical point if b 2 - 3ac f(x) has only one critical point if b 2 - 3ac = 0.f(x) has two critical points if b 2 - 3ac > 0.This is a quadratic equation and we can solve it using the techniques of solving quadratic equations. To find the critical points of a cubic function f(x) = ax 3 + bx 2 + cx + d, we set the first derivative to zero and solve. i.e., a function may have either a maximum or minimum value at the critical point. The critical points of a function are the points where the function changes from either "increasing to decreasing" or "decreasing to increasing". The critical points and inflection points play a crucial role in graphing a cubic function. Therefore, the y-intercept of the function is (0, -4).Ĭritical and Inflection Points of Cubic Function To find the y-intercept of a cubic function, we just substitute x = 0 and solve for y-value.Įxample: To find the y-intercept of f(x) = x 3 - 4x 2 + x - 4, substitute x = 0. Y-Intercept of Cubic FunctionĪ cubic function always has exactly one y-intercept. Therefore, f(x) has only one x-intercept which is (4, 0). ThenĬomplex numbers cannot be the x-intercepts. To find the x-intercept(s) of a cubic function, we just substitute y = 0 (or f(x) = 0) and solve for x-values.Įxample: To find the x-intercept(s) of f(x) = x 3 - 4x 2 + x - 4, substitute f(x) = 0. Thus, it has one or three real roots or x-intercepts. So a function can either have 0 or two complex roots. Since complex roots of any function always occur in pairs, a function will always have 0, 2, 4. As the degree of a cubic function is 3, it can have a maximum of 3 roots. The x-intercepts of a function are also known as roots (or) zeros.
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Let us see how to find the intercepts of a cubic function. Hence a cubic function neither has vertical asymptotes nor has horizontal asymptotes.Īs we know, there are two types of intercepts of a function: x-intercept(s) and y-intercept(s). Since both the domain and range of a cubic function is the set of all real numbers, no values are excluded from either the domain or the range. The asymptotes always correspond to the values that are excluded from the domain and range. Also, if you observe the two examples (in the above figure), all y-values are being covered by the graph, and hence the range of a cubic function is the set of all numbers as well. Since a cubic function y = f(x) is a polynomial function, it is defined for all real values of x and hence its domain is the set of all real numbers (R). Here are some examples of a cubic function.Ī cubic function is an algebraic function as all algebraic functions are polynomial functions. For example, there is only one real number that satisfies x 3 = 0 (which is x = 0) and hence the cubic function f(x) = x 3 has only one real root (the other two roots are complex numbers).
Since a cubic function involves an odd degree polynomial, it has at least one real root. The basic cubic function (which is also known as the parent cube function) is f(x) = x 3. Todd and Mary select Web resources in mathematics for the HRSB website.A cubic function is a polynomial function of degree 3 and is of the form f(x) = ax 3 + bx 2 + cx + d, where a, b, c, and d are real numbers and a ≠ 0. Selected by Todd Adams and Mary O’Neill, Sackville High School, Halifax Regional School Board. QuickMath is an automated service for answering common math problems over the Internet. With elementary, middle and high school sections National Council of Teachers of Mathematics
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High School Servers, online math books, math software to download. Mathematics WWW Virtual Library (From Florida State University) Math Readiness Course from the University of SaskatchewanĮxercises to assess math readiness for university.
Archived math challenges for all grade levels.
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