Sure, let's solve this problem step-by-step.
1. Understanding the question – We are given the function [tex]\( f(x) = x^2 \)[/tex] and a constant [tex]\( k = -1 \)[/tex]. We need to determine which function represents a parabola that opens downward.
2. Analyzing the given function [tex]\( f(x) = x^2 \)[/tex] – This is a standard quadratic function, and its graph is a parabola that opens upwards. This is because the coefficient of [tex]\( x^2 \)[/tex] (which is [tex]\( 1 \)[/tex] here) is positive.
3. Applying the constant [tex]\( k \)[/tex] to the function – We need to multiply [tex]\( f(x) \)[/tex] by [tex]\( k \)[/tex]:
[tex]\[
g(x) = k \cdot f(x) = -1 \cdot x^2 = -x^2
\][/tex]
4. Determining the orientation of the resulting parabola – The function [tex]\( g(x) = -x^2 \)[/tex] represents a parabola. To determine whether the parabola opens upwards or downwards, we inspect the coefficient of [tex]\( x^2 \)[/tex]:
- If the coefficient of [tex]\( x^2 \)[/tex] is negative, the parabola opens downward.
- If the coefficient of [tex]\( x^2 \)[/tex] is positive, the parabola opens upward.
Since the coefficient of [tex]\( x^2 \)[/tex] in [tex]\( g(x) = -x^2 \)[/tex] is [tex]\(-1\)[/tex], which is negative, the parabola opens downward.
Based on this analysis, the function that represents a parabola opening downward is [tex]\( g(x) = -x^2 \)[/tex]. Therefore, the answer to the multiple-choice question is:
[tex]\[
g(x) = -x^2
\][/tex]
This function correctly represents a parabola that opens downward.
