Explore Westonci.ca, the premier Q&A site that helps you find precise answers to your questions, no matter the topic. Get detailed and accurate answers to your questions from a community of experts on our comprehensive Q&A platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
To convert the vertex form of the equation [tex]\( y = \frac{1}{2}(x-4)^2 + 13 \)[/tex] to its standard form, follow these steps:
1. Start with the vertex form:
[tex]\[ y = \frac{1}{2}(x-4)^2 + 13 \][/tex]
2. Expand the squared term [tex]\( (x - 4)^2 \)[/tex]:
[tex]\[ (x-4)^2 = x^2 - 8x + 16 \][/tex]
3. Substitute this expanded form back into the original equation:
[tex]\[ y = \frac{1}{2}(x^2 - 8x + 16) + 13 \][/tex]
4. Distribute the [tex]\(\frac{1}{2}\)[/tex] across the terms inside the parentheses:
[tex]\[ y = \frac{1}{2}x^2 - \frac{1}{2}(8x) + \frac{1}{2}(16) + 13 \][/tex]
Simplifying each term:
[tex]\[ y = \frac{1}{2}x^2 - 4x + 8 + 13 \][/tex]
5. Combine the constant terms:
[tex]\[ y = \frac{1}{2}x^2 - 4x + 21 \][/tex]
This shows that the standard form of the equation is:
[tex]\[ y = \frac{1}{2}x^2 - 4x + 21 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{D. \, y = \frac{1}{2} x^2 - 4 x + 21} \][/tex]
1. Start with the vertex form:
[tex]\[ y = \frac{1}{2}(x-4)^2 + 13 \][/tex]
2. Expand the squared term [tex]\( (x - 4)^2 \)[/tex]:
[tex]\[ (x-4)^2 = x^2 - 8x + 16 \][/tex]
3. Substitute this expanded form back into the original equation:
[tex]\[ y = \frac{1}{2}(x^2 - 8x + 16) + 13 \][/tex]
4. Distribute the [tex]\(\frac{1}{2}\)[/tex] across the terms inside the parentheses:
[tex]\[ y = \frac{1}{2}x^2 - \frac{1}{2}(8x) + \frac{1}{2}(16) + 13 \][/tex]
Simplifying each term:
[tex]\[ y = \frac{1}{2}x^2 - 4x + 8 + 13 \][/tex]
5. Combine the constant terms:
[tex]\[ y = \frac{1}{2}x^2 - 4x + 21 \][/tex]
This shows that the standard form of the equation is:
[tex]\[ y = \frac{1}{2}x^2 - 4x + 21 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{D. \, y = \frac{1}{2} x^2 - 4 x + 21} \][/tex]
We appreciate your time. Please come back anytime for the latest information and answers to your questions. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.