The prompt says "Graph each function. Determine the interval[s] for which the function is increasing and the interval[s] for which the function is decreasing."
The problems that are given in this prompt are:
1. y= x^3+3x^2-9x
2. f(x)=1/(x+1)-4
3. g(x)=(x^2+5)/(x-2)
I need mega help..
how far along are you? Can you tell when a function is increasing or decreasing?
First step: try sketching the graphs as asked. You've had a couple of years of algebra now, so you should be able to find roots, asymptotes, etc.
You can also visit
rechneronline.de/function-graphs/
for some excellent help. Play around there and get a feel for how these functions work.
I got the first one. I guess what I don't understand is the Asymptotes. Do I include them in my answer?
Cause, I feel like they have to be included.
Sure, I can help you with these problems. To determine intervals of increase and decrease, we need to understand the concept of the derivative of a function.
1. For the function y = x^3 + 3x^2 - 9x:
a) To graph the function, you can start by plotting some points. Choose various x-values and substitute them into the equation to find the corresponding y-values. Plot these points on a graph and connect them to get a smooth curve.
b) To determine intervals of increase and decrease, we need to find the derivative of the function. Taking the derivative of y with respect to x, denoted as dy/dx, will give us the slope of the tangent line at any point on the graph.
c) Set the derivative equal to zero and solve for x to find the critical points. The critical points are where the tangent line is horizontal or has zero slope.
d) Using these critical points, we can determine the intervals where the function is increasing or decreasing. If the derivative is positive in an interval, the function is increasing, and if the derivative is negative, the function is decreasing.
2. For the function f(x) = 1/(x+1) - 4:
a) Start by graphing the function by plotting points or using a graphing calculator.
b) Find the derivative of f(x) using the same steps as before.
c) Set the derivative equal to zero and solve for x to find the critical points.
d) Use the critical points to determine the intervals of increase and decrease.
3. For the function g(x) = (x^2 + 5)/(x - 2):
a) Graph the function by plotting points or using a graphing calculator.
b) Find the derivative of g(x) using the steps mentioned earlier.
c) Identify the critical points by setting the derivative equal to zero and solving for x.
d) Determine the intervals of increase and decrease using the critical points.
Remember, the derivative gives us information about the slope of the function at different points, which helps in determining where the function is increasing or decreasing.