write a function g that has a y-intercept of -2 and a transformation of f(x)=-.25(2x^2-3)(x+2)

To write the function \( g(x) \) based on the transformation of \( f(x) = -0.25(2x^2 - 3)(x + 2) \) and ensuring that the function has a y-intercept of \(-2\), follow these steps:

1. **Start with the given function**:
   \[
   f(x) = -0.25(2x^2 - 3)(x + 2)
   \]

2. **To adjust the y-intercept**:  
   The y-intercept is where \( x = 0 \). To adjust this, you can add or subtract a constant (vertical shift) to ensure that \( g(0) = -2 \).

3. **Find the y-intercept of \( f(x) \)**:
   Substitute \( x = 0 \) into \( f(x) \):
   \[
   f(0) = -0.25(2(0)^2 - 3)(0 + 2)
   = -0.25(-3)(2)
   = 1.5
   \]
   So, the y-intercept of \( f(x) \) is \( 1.5 \).

4. **Shift the function vertically**:  
   To shift the function down by \( 1.5 + 2 = 3.5 \) (to move the y-intercept from \( 1.5 \) to \( -2 \)), subtract \( 3.5 \) from the function.

Thus, the new function \( g(x) \) is:
\[
g(x) = -0.25(2x^2 - 3)(x + 2) - 3.5
\]

### Final Answer:
The function \( g(x) \) is:
\[
g(x) = -0.25(2x^2 - 3)(x + 2) - 3.5
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the vertex of the quadratic component in \( f(x) \)?
2. How would you find the roots of \( g(x) \)?
3. How does the vertical shift affect the graph of a function?
4. What is the domain and range of \( g(x) \)?
5. Can you describe the end behavior of \( g(x) \)?

**Tip**: When transforming functions, always check the effect on critical points like the y-intercept and roots to ensure accuracy in shifts and stretches.

Learn how to transform the function f(x) = -0.25(2x^2 - 3)(x + 2) to create a new function g(x) with a y-intercept of -2. This problem demonstrates key concepts in function transformations, including vertical shifts, and is suitable for high school students in Grades 10-12.

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