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.