https://file.aiquickdraw.com/mathbot/file/1723830227570f07505ip.jpg

To create an animated design on Desmos that meets the criteria provided in your image, follow these steps:

### Step-by-Step Guide:

1. **Polynomial Function (Degree 3 or Higher):**
   - Example: \( f_1(x) = x^3 - 3x + 2 \)
   
2. **Exponential Function:**
   - Example: \( f_2(x) = 2^x \)
   
3. **Logarithmic Function:**
   - Example: \( f_3(x) = \log_2(x) \)
   
4. **Trigonometric Function:**
   - Example: \( f_4(x) = \sin(x) \)
   
5. **Rational Function:**
   - Example: \( f_5(x) = \frac{1}{x-2} \)

6. **Sum or Difference Function with Local Maximum or Minimum:**
   - Combine two functions from different categories, such as trigonometric and polynomial.
   - Example: \( f_6(x) = \sin(x) + x^3 - 3x \)
   
7. **Product Function with \(x\)-Intercepts:**
   - Choose functions from different categories and multiply them.
   - Example: \( f_7(x) = \sin(x) \times \frac{1}{x-2} \)
   
8. **Quotient Function:**
   - Choose functions from different categories and divide them.
   - Example: \( f_8(x) = \frac{\sin(x)}{x^3 - 3x + 2} \)

9. **Composite Function:**
   - The inner and outer functions must be from different categories.
   - Example: \( f_9(x) = \sin(x^3 - 3x) \)

### Instructions for Desmos:
1. **Open Desmos:**
   - Go to [Desmos Graphing Calculator](https://www.desmos.com/calculator).

2. **Enter Each Function:**
   - Enter the equations \( f_1(x) \) to \( f_9(x) \) in separate lines.

3. **Animate the Design:**
   - Use Desmos's animation feature by adding a slider. For example, set \( a = \sin(t) \), and replace a constant in any of the functions with \( a \).
   - Example: \( f_1(x) = a \times (x^3 - 3x + 2) \)
   - Set \( t \) to animate over time, and adjust the speed for the desired effect.

4. **Ensure Functions Meet the Criteria:**
   - Confirm that all functions fulfill the requirements (e.g., different categories, not simplifying to linear or quadratic).

### Visualization:

After entering these functions into Desmos, adjust the view and scale to create a visually appealing animated design. You can customize colors, thickness, and add sliders to control the animation.

Would you like more details on any of these steps, or do you have any specific questions?

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### Relative Questions:
1. How do you create a slider in Desmos for animation purposes?
2. Can you explain how to identify a local maximum or minimum in a function?
3. How do you determine the category of a function in a complex expression?
4. What are the effects of varying parameters in an exponential function?
5. How can you combine multiple functions into a single composite function?
6. How do you find the \(x\)-intercepts of a product function in Desmos?
7. What are some visual tips for optimizing graph appearance in Desmos?
8. How do you ensure that a function does not simplify to a linear or quadratic?

### Tip:
In Desmos, using sliders with functions can create dynamic and engaging visualizations, making it easier to explore the behavior of complex functions interactively.

Learn how to create dynamic animated designs on Desmos using polynomial, exponential, logarithmic, trigonometric, rational, and composite functions. Follow a step-by-step guide with examples and instructions for optimizing your visualizations.

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