how do you graph the function?
f(x)=2(x+2)^2(x+4)^2
visualize the function as
f(x) = 2(x+2)(x+2)(x+4)(x+4)
which would make it a fourth degree function.
Fourth degree functions tend to look like a "W" in their general form, anyway they "rise" into the first and second quadrants.
notice there are double roots at x=-2 and x=-4, so the curve 'touches' the x-axis at -2 and -4 without crossing
What does it do between -2 and -4?
Try x=-3, f(-3) = 2
try f(0), you get f(0) = 128
Can you see the "W" ?
To graph the function f(x) = 2(x+2)^2(x+4)^2, we can follow these steps:
1. Determine the key points:
- Find the x-intercepts by setting f(x) = 0 and solving for x. In this case, the function will be equal to zero when either (x+2)^2 = 0 or (x+4)^2 = 0. Solve these equations to obtain the x-intercepts.
- Find the y-intercept by substituting x = 0 into the function and calculating f(0).
2. Analyze the behavior near x-intercepts:
- Determine the end behavior as x approaches positive or negative infinity. This will help establish the shape of the graph.
3. Examine the symmetry of the function:
- Determine if the function is symmetric with respect to the y-axis or origin by substituting (-x) for (x) in the function and simplifying.
4. Sketch the graph:
- Plot the key points obtained from the steps above, including x-intercepts and the y-intercept.
- Make use of symmetry, if any, to ensure accuracy in graphing.
- Determine the shape of the graph by analyzing the end behavior.
- Connect the points smoothly to create the graph of the function.
Note: It is always beneficial to use graphing software or tools to quickly determine the shape and features of complicated functions.