Get the answers you need at Westonci.ca, where our expert community is always ready to help with accurate information. Get immediate and reliable answers to your questions from a community of experienced professionals on our platform. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
To determine the minimum value of the function [tex]\( y = -2 + 5 \sin \left( \frac{\pi}{12} (x - 2) \right) \)[/tex], you need to analyze the behavior of the sine component within the function.
1. Consider the sine function: [tex]\(\sin \theta\)[/tex]. The sine function, [tex]\(\sin \theta\)[/tex], oscillates between [tex]\(-1\)[/tex] and [tex]\(1\)[/tex].
2. In the expression [tex]\( 5 \sin \left( \frac{\pi}{12} (x - 2) \right) \)[/tex], the sine function [tex]\(\sin \left( \frac{\pi}{12} (x - 2) \right)\)[/tex] will also oscillate between [tex]\(-1\)[/tex] and [tex]\(1\)[/tex], regardless of the argument [tex]\(\frac{\pi}{12} (x - 2)\)[/tex].
3. To find the maximum and minimum values of [tex]\(5 \sin \left( \frac{\pi}{12} (x - 2) \right)\)[/tex], we multiply the range [tex]\([-1, 1]\)[/tex] of [tex]\(\sin \theta\)[/tex] by the coefficient [tex]\(5\)[/tex]:
[tex]\[ 5 \times -1 = -5 \quad \text{and} \quad 5 \times 1 = 5 \][/tex]
Hence, the expression [tex]\(5 \sin \left( \frac{\pi}{12} (x - 2) \right)\)[/tex] oscillates between [tex]\(-5\)[/tex] and [tex]\(5\)[/tex].
4. Now, consider the full expression [tex]\( y = -2 + 5 \sin \left( \frac{\pi}{12} (x - 2) \right) \)[/tex]. To find the minimum value of [tex]\(y\)[/tex], we need the smallest value of the term [tex]\( 5 \sin \left( \frac{\pi}{12} (x - 2) \right) \)[/tex].
- The minimum value of [tex]\( 5 \sin \left( \frac{\pi}{12} (x - 2) \right) \)[/tex] is [tex]\(-5\)[/tex].
5. Substitute this minimum value back into the expression for [tex]\( y \)[/tex]:
[tex]\[ y = -2 + (-5) \][/tex]
6. Simplify the expression:
[tex]\[ y = -2 - 5 = -7 \][/tex]
Therefore, the minimum value of the function [tex]\( y = -2 + 5 \sin \left( \frac{\pi}{12} (x - 2) \right) \)[/tex] is [tex]\(\boxed{-7}\)[/tex].
1. Consider the sine function: [tex]\(\sin \theta\)[/tex]. The sine function, [tex]\(\sin \theta\)[/tex], oscillates between [tex]\(-1\)[/tex] and [tex]\(1\)[/tex].
2. In the expression [tex]\( 5 \sin \left( \frac{\pi}{12} (x - 2) \right) \)[/tex], the sine function [tex]\(\sin \left( \frac{\pi}{12} (x - 2) \right)\)[/tex] will also oscillate between [tex]\(-1\)[/tex] and [tex]\(1\)[/tex], regardless of the argument [tex]\(\frac{\pi}{12} (x - 2)\)[/tex].
3. To find the maximum and minimum values of [tex]\(5 \sin \left( \frac{\pi}{12} (x - 2) \right)\)[/tex], we multiply the range [tex]\([-1, 1]\)[/tex] of [tex]\(\sin \theta\)[/tex] by the coefficient [tex]\(5\)[/tex]:
[tex]\[ 5 \times -1 = -5 \quad \text{and} \quad 5 \times 1 = 5 \][/tex]
Hence, the expression [tex]\(5 \sin \left( \frac{\pi}{12} (x - 2) \right)\)[/tex] oscillates between [tex]\(-5\)[/tex] and [tex]\(5\)[/tex].
4. Now, consider the full expression [tex]\( y = -2 + 5 \sin \left( \frac{\pi}{12} (x - 2) \right) \)[/tex]. To find the minimum value of [tex]\(y\)[/tex], we need the smallest value of the term [tex]\( 5 \sin \left( \frac{\pi}{12} (x - 2) \right) \)[/tex].
- The minimum value of [tex]\( 5 \sin \left( \frac{\pi}{12} (x - 2) \right) \)[/tex] is [tex]\(-5\)[/tex].
5. Substitute this minimum value back into the expression for [tex]\( y \)[/tex]:
[tex]\[ y = -2 + (-5) \][/tex]
6. Simplify the expression:
[tex]\[ y = -2 - 5 = -7 \][/tex]
Therefore, the minimum value of the function [tex]\( y = -2 + 5 \sin \left( \frac{\pi}{12} (x - 2) \right) \)[/tex] is [tex]\(\boxed{-7}\)[/tex].
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.