Cubic Functions: Graphs, Equations, and Intercepts Explained
Learn about cubic functions, how to plot and draw their graphs, find their equations using intercepts, and understand their characteristics such as points of inflection. Explore examples and practical applications in mathematics.
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
You are allowed to download the files provided on this website for personal or commercial use, subject to the condition that they are used lawfully. All files are the property of their respective owners.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.
E N D
Presentation Transcript
9 June 2025 Cubic function LO: To plot and draw graphs with equation: y = Ax3 + Bx2 + Cx + D www.mathssupport.org www.mathssupport.org
Cubic functions A cubic function has the form f(x) =ax3 + bx2 + cx + d where a 0 and a, b, c, and d are constants. Here are examples of three cubic functions: y = -3x2 x3 y = x3 4x y = x3 + 2x2 www.mathssupport.org www.mathssupport.org
Cubic functions for values of x between 3 and 3. Plot the graph of y = x3 3 2 1 0 1 2 3 x y = x3 27 8 1 0 1 8 27 y y The points given in the table are plotted and joined together 40 30 20 Notice that as x increases,y increases rapidly 10 x 3 2 1 0 1 2 3 The curveis flat at (0,0). This point is called a point of inflection. This is the point of rotational symmetry 10 20 30 www.mathssupport.org www.mathssupport.org
Using axis intercepts Use the axes intercepts to sketch the graph of this function f(x) = (x + 3)(x + 1)(x 2) x-intercepts occur when y = 0 ( 3, 0) ( 1, 0) (2, 0) y-intercepts occur when x = 0 f(0) = (0 + 3)(0 + 1)(0 2) f(0) = 6 Asa > 0 the graph has this shape y y x 3 2 1 0 1 2 3 and the points are then joined together with a smooth curve. The shape of this graph is characteristic of a cubic function. 6 www.mathssupport.org www.mathssupport.org
Using axis intercepts Use the axes intercepts to sketch the graph of this function f(x) = x (x + 3)2 x-intercepts occur when y = 0 (0, 0) ( 3, 0) y-intercepts occur when x = 0 f(0) = 0(0 + 3)2 f(0) = 0 y y x As a < 0 the graph has this shape 3 2 1 0 1 2 3 and the points are then joined together with a smooth curve. www.mathssupport.org www.mathssupport.org
Finding a cubic function Find the equation of the cubic function with this graph The x-intercepts are 1, 2 , and 4 y y f(x) = a(x + 1)(x 2)(x 4) When x = 0 y = 8 x 4 2 1 0 8 = a(0 + 1)(0 2)(0 4) 8 = a(1)(2)(4) 8 = 8a a = 1 8 So, f(x) = (x + 1)(x 2)(x 4) www.mathssupport.org www.mathssupport.org
Finding a cubic function Find the equation of the cubic function with this graph The x-intercepts are -3, and touches at 1 f(x) = a(x + 3)(x 1)2 y y When x = 0 y = 6 6 6 = a(0 + 3)(0 1)2 6 = a(3)(1)2 x 3 1 0 6 = 3a a = 2 So, f(x) = 2(x + 3)(x 1)2 www.mathssupport.org www.mathssupport.org
Finding a cubic function Find the equation of the cubic function which has x-intercepts 1 and 3, y-intercept 9 and passes through ( 1, 8) The x-intercepts are 1 and 3, so (x 1) and (x 3) are linear factors We suppose the third factor is (ax + b) So, f(x) = (ax + b)(x 1)(x 3) When f( 1) = 8 When f(0) = 9 8 = ( 1a + 3)( 1 1)( 1 3) 8 = (3 a)( 2)( 4) 8 = (3 a)(8) 1 = 3 a a = 2 9 = (0a + b)(0 1)(0 3) 9 = b(-1)(-3) 9 = 3b b = 3 So, f(x) = (2x + 3)(x 1)(x 3) f(x) = 2x3 5x2 6x + 9 www.mathssupport.org www.mathssupport.org
Thank you for using resources from A close up of a cage Description automatically generated For more resources visit our website https://www.mathssupport.org If you have a special request, drop us an email info@mathssupport.org Get 20% off in your next purchase from our website, just use this code when checkout: MSUPPORT_20 www.mathssupport.org www.mathssupport.org
