How do you graph f(x) = -x(2x+5)^2(3x-10)?

1 Answer
Jul 2, 2017

There isn't a way to do this accurately, but if you just want to sketch the graph, read below for the quite complicated solution.

Explanation:

Since f(x)=0 when either -x, 2x+5, or 3x-10 are equal to zero, f(x) has zeros at x=0, -5/2, and 10/3.

So now we have 3 points to work with. To find out what happens at each point, we can just look at the different parts of the equation.

To find out what happens around x=0, just look at the -x part of the function. This is linear, so the function passes the point (0, 0) in a straight line.

To find out what happens around x=-5/2, just look at the (2x+5)^2 part of the function. This is quadratic, and it's a perfect square, so there's a double root at x=-5/2. This means that the function just touches the x-axis at this point.

To find out what happens around x=10/3, just look at the 3x-10 part of the function. This is linear, so the function just passes through the point (10/3, 0) in a straight line.

We can also find out what happens at the two extreme ends of the function (the end behavior of the function). If x is really big (approaching \infty), then f(x) would be really small (approaching -\infty), since every part of the function would return a positive number, and there's a negative sign in front of everything. If x is really small (approaching -\infty), then f(x) would also be really small (approaching -\infty).

With all of this information, we can sketch the graph of the function:

The function starts out on the left side being really negative, and then goes towards the x-axis, but never actually crossing the x-axis at x=-5/2. It touches the x-axis, and goes back up (like the function y=x^2). At x=0, it passes through the point (0 ,0 ) in a straight line (like the function y=-x). It then goes back towards the x-axis at x=10/3, passing through the point (0, 10/3) in a straight line (like the equation y=x).

You can check with the below graph:
graph{-x(2x+5)^2(3x-10) [-5, 5, -700, 700]}

OR, if you want a more simple way to graph the function, just plug it into your graphing calculator or WolframAlpha or any graphing utility.