So I'm doing polynomial functions right now and it's talking about changes in direct of the graphs. Just wondering, is it possible to have more than 3-4 changes in direction on a single line? I've messed around trying to get a really squiggly, continuous line with one function on Desmos and nothing works.
increasing the order of the highest order term
should result in more "squiggles"
y = a x ... line
y = a x^2 ... parabola ... one direction change
y = a x^3 ... two changes
etc.
y = (x-c1)(x-c2)...(x-cn)
with n factors has n-1 changes in direction
Yes, it is possible to have more than 3-4 changes in direction on a single line. However, the specific number of changes in direction depends on the degree of the polynomial function.
To create a polynomial function with multiple changes in direction, you need to include alternating terms with positive and negative coefficients for the powers of x. The number of changes in direction corresponds to the number of times the sign of the coefficient changes.
For example, a quadratic function (degree 2) can have only one change in direction since it has one change of sign in its coefficient. A cubic function (degree 3) can have up to two changes in direction, and a quartic function (degree 4) can have up to three changes in direction.
To create a more squiggly, continuous line with a single polynomial function in Desmos, you can try increasing the degree of the polynomial. This will allow for more changes in direction. You can experiment by adding more terms to the polynomial function until you achieve the desired result.
Keep in mind that the more changes in direction, the more complex the graph becomes, and it may be difficult to visualize in a simple graphing calculator like Desmos.