Westonci.ca is the premier destination for reliable answers to your questions, brought to you by a community of experts. Join our platform to get reliable answers to your questions from a knowledgeable community of experts. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
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.
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.
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.