Sketch the graph of 1/12(x+2)^2(x-3)^2 and make sure the graph shows all intercepts and exhibits the proper end behavior.
I can sketch the graph but how do I do the second part? Thanks
What does the function do at x= inf, or x= -inf. That is the end behaviour.
To determine the end behavior of a function, you can analyze the behavior of the function as x approaches positive infinity and negative infinity. Here's how you can do it step-by-step:
1. Determine the degree of the polynomial. In this case, the function is a product of two quadratic terms, (x+2)^2 and (x-3)^2, so the highest power of x is 2.
2. Since the degree of the polynomial is even, the end behavior of the function will be the same on both sides of the x-axis. In other words, the behavior as x approaches positive infinity will be the same as when x approaches negative infinity.
3. Analyze the leading coefficient, the coefficient of the highest power of x. In this case, the leading coefficient is 1/12.
4. Because the leading coefficient is positive, as x approaches both positive infinity and negative infinity, the function will also approach positive infinity.
Therefore, the proper end behavior of the function f(x) = (1/12)(x+2)^2(x-3)^2 is that it increases without bound as x approaches both positive infinity and negative infinity.
Now, when you sketch the graph, make sure to show all intercepts (x and y-intercepts) and exhibit the proper end behavior by extending the graph in the correct direction as x approaches infinity.