f(x)=-|x^2|
y=1/2[f(x)]
what is the graph of y

Given the function \( f(x) = -|x^2| \), let's first analyze the behavior of \( f(x) \).

### Step 1: Understanding \( f(x) = -|x^2| \)
- The expression \( |x^2| \) is simply \( x^2 \), since \( x^2 \) is always non-negative.
- Thus, \( f(x) = -x^2 \).

So, the function \( f(x) = -|x^2| \) is equivalent to \( f(x) = -x^2 \), which represents a downward-facing parabola centered at the origin.

### Step 2: Now, consider \( y = \frac{1}{2} f(x) \)
- Since \( f(x) = -x^2 \), the equation for \( y \) becomes:
  \[
  y = \frac{1}{2}(-x^2) = -\frac{1}{2}x^2
  \]

This is also a parabola, but it is a scaled version of the original \( f(x) = -x^2 \). The factor \( \frac{1}{2} \) compresses the parabola vertically, making it "wider."

### Characteristics of the graph \( y = -\frac{1}{2}x^2 \):
- **Vertex**: The vertex is at \( (0, 0) \), the same as the vertex of the parent function \( -x^2 \).
- **Direction**: The parabola opens downward.
- **Shape**: The factor \( \frac{1}{2} \) makes the parabola wider than the graph of \( f(x) = -x^2 \), as it reduces the steepness of the curve.

Would you like more details or have any specific questions?

Here are 5 relative questions:
1. How does a vertical compression affect the graph of a function?
2. What is the difference between vertical compression and vertical stretch in parabolas?
3. How can the vertex form of a parabola help describe transformations?
4. How do you determine the axis of symmetry for a parabola?
5. What happens to the graph of a quadratic function when you change its leading coefficient?

**Tip**: When scaling a function, multiplying by a factor less than 1 compresses it vertically, while multiplying by a factor greater than 1 stretches it vertically.

f(x)=-|x^2|, y=1/2[f(x)] what is the graph of y

Explore how to graph the function y = -1/2x^2 by analyzing the transformation of the quadratic function f(x) = -|x^2|. Learn about vertical compression and the properties of parabolas, including the vertex, direction, and shape of the graph. Suitable for students in Grades 9-12.

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