![]() ![]() In graph transformations, however, all transformations done directly to x take the opposite direction expected. This may seem counterintuitive because, typically, negative numbers represent left movement and positive numbers represent right movement. In the given function, we subtract 2 from x, which represents a vertex shift two units to the right. In this case, we need to remember that all numbers added to the x-term of the function represent a horizontal shift while all numbers added to the function as a whole represent a vertical shift. ![]() Example 2 SolutionĪgain, we will use the parent function x 3 to find the graph of the given function. This point is also the only x-intercept or y-intercept in the function. Thus, the function -x 3 is simply the function x 3 reflected over the x-axis. If we multiply a cubic function by a negative number, it reflects the function over the x-axis. The only difference between the given function and the parent function is the presence of a negative sign. This section will go over how to graph simple examples of cubic functions without using derivatives. This is a rather long formula, so many people rely on calculators to find the zeroes of cubic functions that cannot easily be factored. From the initial form of the function, however, we can see that this function will be equal to 0 when x=0, x=1, or x=-1. For example, the function x(x-1)(x+1) simplifies to x 3-x. Unlike quadratic functions, cubic functions will always have at least one real solution. In the parent function, the y-intercept and the vertex are one and the same. To find it, you simply find the point f(0). If this number, a, is negative, it flips the graph upside down as shown.Īs with quadratic functions and linear functions, the y-intercept is the point where x=0. For example 0.5x 3 compresses the function, while 2x 3 widens it. ReflectionĪs before, if we multiply the cubed function by a number a, we can change the stretch of the graph. For example, the function x 3+1 is the cubic function shifted one unit up. To shift this function up or down, we can add or subtract numbers after the cubed part of the function. For example, the function (x-1) 3 is the cubic function shifted one unit to the right. To shift this vertex to the left or to the right, we can add or subtract numbers to the cubed part of the function. In the parent function, this point is the origin. The vertex of the cubic function is the point where the function changes directions. It has a shape that looks like two halves of parabolas that point in opposite directions have been pasted together. The parent function, x 3, goes through the origin. Here, we will focus on how we can use graph transformations to find the shape and key points of a cubic function. This will be covered in greater depth, however, in calculus sections about using the derivative. Then, we can use the key points of this function to figure out where the key points of the cubic function are. In particular, we can find the derivative of the cubic function, which will be a quadratic function. There are methods from calculus that make it easy to find the local extrema. Graphing cubic functions will also require a decent amount of familiarity with algebra and algebraic manipulation of equations.īefore graphing a cubic function, it is important that we familiarize ourselves with the parent function, y=x 3. In particular, we can use the basic shape of a cubic graph to help us create models of more complicated cubic functions.īefore learning to graph cubic functions, it is helpful to review graph transformations, coordinate geometry, and graphing quadratic functions. Graphing cubic functions is similar to graphing quadratic functions in some ways. ![]() Graphing cubic functions gives a two-dimensional model of functions where x is raised to the third power. Graphing Cubic Function – Explanation and Examples
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