To determine the correct transformation applied to the function [tex]\(f(x) = x^2\)[/tex] to obtain the function [tex]\(g(x)\)[/tex], we need to analyze the changes step-by-step.
1. The original function is [tex]\(f(x) = x^2\)[/tex].
2. The given form of the transformed function is [tex]\(g(x) = \boxed{} -2x^2 - 3\)[/tex].
Now let's examine the changes:
- Multiplication by -2:
[tex]\( -2x^2 \)[/tex] reflects the graph of [tex]\(x^2\)[/tex] over the x-axis and stretches it vertically by a factor of 2. The negative sign indicates the reflection, while the 2 indicates the stretch.
- Subtraction of 3:
[tex]\(-3\)[/tex] translates the graph downward by 3 units. This moves every point on the graph down by 3 units.
Combining these transformations, the function [tex]\(g(x)\)[/tex] can be written as:
[tex]\[ g(x) = -2x^2 - 3 \][/tex]
Thus, the correct transformation applied to [tex]\(f(x) = x^2\)[/tex] to get [tex]\(g(x)\)[/tex] is:
[tex]\[ \boxed{} = -2 \][/tex]
\x^2 - 3\)
