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Graphing Cubic Functions

Graphing Cubic Functions

A cubic function will have an `x^3` term. An example of a cubic function is `2x^3 + 8x^2 - 2x - 8`.

A cubic function will have either one, two or three distinct solutions, where the function crosses the x-axis. This can be determined by factorising the function: the above example factorises to `(x + 4)(2x - 2)(x + 1)`, giving roots of -4, 1 and -1. The function `y = (x - 4)^3` has one distinct solution (+4).

The intersection with the `y`-axis can be obtained by identifying the number part of the function.

Example 1

Sketch the graph of `y = x^3-3x^2-x+3`. Mark the points of intersection with both the `x` and `y`-axis.

Hint: the function factorises to `(x-3)(x-1)(x+1)`.

Substitute a value into each bracket to make that bracket zero eg for the first bracket, set `x`=3 to make `(x-3)=0` therefore `x` is a solution. Similarly for the other two brackets. Graph of f(x)=x<sup>3-3x</sup>2-x+3

Answer: The roots of the equation are 3, 1 and -1.

Example 2

How many distinct solutions does `y = x^3 - 1` have?

The only solution to `x^3-1` is when `x`=1

Graph of f(x)=x<sup>3-1

Answer: 1

See also Cubes