Graph of f(x) = (x^3 - 5x ^2 + 6x)/(x^2 - x- 2). Can anyone please post a picture of this or something?
go to wolframalpha dot com and put in (x^3 - 5x ^2 + 6x)/(x^2 - x- 2), it will show you a plot
We don't draw graphs here. I suggest you simplify it first by factoring.
It should be hard for you to graph this yourself. Just plot a dozen points or so that you can calculate easily.
(x^3 - 5x ^2 + 6x)/(x^2 - x- 2)
= x(x-6)(x+1))/[(x-2)(x+1)]
= x*(x-6)/(x-2)
When x = -20, y = -23.6
When x = -15, y = -18.5
when x = -10, y = -13.3
When x = -5, y = -7.85
When x = 0, y = 0
When x = 2 (left side) y = +infinity
When x = 2 (right side)y = -infinity
When x = 6, y = 0
When x = 10, y = 5
When x = 20, y = 15.5
The graph will have a vertical asymptote at x=2, and for very large positive and negative x, it will be asymptotic to the line y = x.
To graph the function f(x) = (x^3 - 5x^2 + 6x)/(x^2 - x - 2), you will need to follow these steps:
1. Determine the x-intercepts: Set the numerator equal to zero and solve for x. The values of x that make the numerator equal to zero are the x-intercepts of the graph.
(x^3 - 5x^2 + 6x) = 0
x(x^2 - 5x + 6) = 0
x(x - 2)(x - 3) = 0
So, the x-intercepts are x = 0, x = 2, and x = 3.
2. Determine the vertical asymptotes: Set the denominator equal to zero and solve for x. The values of x that make the denominator equal to zero are the vertical asymptotes of the graph.
(x^2 - x - 2) = 0
(x - 2)(x + 1) = 0
So, the vertical asymptotes are x = 2 and x = -1.
3. Determine the horizontal asymptote: Divide the numerator by the denominator using long division or synthetic division. The quotient will give you the equation for the horizontal asymptote.
(x^3 - 5x^2 + 6x) ÷ (x^2 - x - 2) = (x - 3)
Therefore, the horizontal asymptote is y = (x - 3).
4. Find additional points: Choose a few x-values to find y-values to plot additional points on the graph. You can choose values such as x = -2, x = 1, and x = 4, for example.
Calculate the corresponding y-values by substituting the chosen x-values into the original function.
5. Plot points and sketch the graph: Use the obtained x- and y-values to plot the points on a coordinate plane. Connect the points smoothly to create a curve that represents the graph of the function.
Since I am an AI bot and cannot directly post pictures, I encourage you to use various graphing tools available online or through graphing calculators to plot the graph of f(x) = (x^3 - 5x^2 + 6x)/(x^2 - x - 2). These tools allow you to input the function and visualize the graph easily. Simply input the function, and the graphing tool will generate the graph for you.