Here’s a step-by-step process for creating a polynomial function meeting your requirements, along with an explanation of how to plot it on Desmos.
Step 1: Create the Polynomial Function
To fulfill the requirements of having 3 turning points and the specified zeros of -2 and 4, you can create a polynomial function of at least degree 4 (since a polynomial of degree n can have at most n-1 turning points).
A suitable polynomial function with the specified zeros might look like this:
\[ f(x) = (x + 2)(x - 4)(x - 1)(x - 3) \]
This function has:
- Zeros at: x = -2, x = 4 (as well as additional zeros at x = 1 and x = 3)
- Degree of 4, allowing for exactly 3 turning points.
Step 2: Expand the Polynomial Function
To get a visual in Desmos and understand its shape better, you may want to expand the function:
- Start with \((x + 2)(x - 4) = x^2 - 2x - 8\)
- Multiply this with \((x - 1)(x - 3) = x^2 - 4x + 3\)
- Multiply these two quadratic results together.
This gives:
\[ f(x) = (x^2 - 2x - 8)(x^2 - 4x + 3) \]
Step 3: Graph Your Function in Desmos
- Open Desmos Graphing Calculator.
- Enter the function as is (or enter the expanded version if preferred).
- Adjust the view to clearly see the turning points and zeros.
Final Function
For the sake of simplicity, you can use the following function directly:
\[ f(x) = (x + 2)(x - 4)(x - 1)(x - 3) \]
Step 4: Take a Screenshot
Once you have the graph of the function in Desmos, zoom in/out to ensure all turning points and zeros are visible, then take a screenshot for submission.
Note:
You can always tweak this function further if you desire other characteristics, like altering the leading coefficient for steeper/shallower curves.
Make sure to include the screenshot with the graph showing 3 turning points and zeros at -2 and 4!