Get the answers you need at Westonci.ca, where our expert community is always ready to help with accurate information. Connect with a community of experts ready to help you find solutions to your questions quickly and accurately. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
Let's analyze the function [tex]\( y = -\frac{1}{2} \sqrt{x+3} + 2 \)[/tex] step by step to understand how to graph it:
1. Identify and Shift:
- The basic form of the radical function is [tex]\( \sqrt{x} \)[/tex].
- Notice the term [tex]\( \sqrt{x+3} \)[/tex]. This indicates a horizontal shift to the left by 3 units.
2. Vertical Scaling and Reflection:
- The function has a multiplier of [tex]\(-\frac{1}{2}\)[/tex] outside the square root.
- The [tex]\(-1/2\)[/tex] factor means we scale the square root function by [tex]\(-1/2\)[/tex], which includes two transformations:
- Reflection: The negative sign reflects the function across the x-axis.
- Vertical Compression: The 1/2 factor compresses the function vertically by half.
3. Vertical Shift Upward:
- There is an additional [tex]\( +2 \)[/tex] outside the square root, which shifts the graph up by 2 units.
To summarize the steps:
- Start from the basic function [tex]\( y = \sqrt{x} \)[/tex].
- Shift it left by 3 to get [tex]\( y = \sqrt{x+3} \)[/tex].
- Reflect and compress vertically by 1/2: [tex]\( y = -\frac{1}{2} \sqrt{x+3} \)[/tex].
- Shift the graph up by 2 units to get the final function: [tex]\( y = -\frac{1}{2} \sqrt{x+3} + 2 \)[/tex].
### Key Points:
To graph this function, it's helpful to calculate and plot specific points by substituting values for [tex]\( x \)[/tex] and solving for [tex]\( y \)[/tex].
1. Determine the domain:
- The expression under the square root (inside √) must be non-negative.
- [tex]\( x + 3 \geq 0 \)[/tex]
- [tex]\( x \geq -3 \)[/tex]
- Thus, the domain is [tex]\( x \geq -3 \)[/tex].
2. Choose key points:
- When [tex]\( x = -3 \)[/tex]:
[tex]\[ y = -\frac{1}{2} \sqrt{-3 + 3} + 2 = -\frac{1}{2} \sqrt{0} + 2 = 2 \][/tex]
- When [tex]\( x = 1 \)[/tex] (a simple value choice):
[tex]\[ y = -\frac{1}{2} \sqrt{1 + 3} + 2 = -\frac{1}{2} \sqrt{4} + 2 = -\frac{1}{2} \cdot 2 + 2 = -1 + 2 = 1 \][/tex]
3. Plot the points and shape:
- Point (-3, 2)
- Point (1, 1)
- Reflecting the downward trend caused by the negative coefficient.
### Graph:
1. Start from point [tex]\((-3, 2)\)[/tex].
2. Draw the curve reflecting the pattern and using additional intermediate values if required.
3. The graph starts from [tex]\((-3, 2)\)[/tex] and decreases as [tex]\( x \)[/tex] increases, smoothly resembling a vertically compressed, reflected half-parabola.
To conclude, by plotting several points and smoothly connecting them, we can draw the correct graph of the function [tex]\( y = -\frac{1}{2} \sqrt{x+3} + 2 \)[/tex]. The graph starts at [tex]\((-3, 2)\)[/tex] and stretches to the right, moving downward, flattened one-half compared to [tex]\( y = -\sqrt{x} \)[/tex].
1. Identify and Shift:
- The basic form of the radical function is [tex]\( \sqrt{x} \)[/tex].
- Notice the term [tex]\( \sqrt{x+3} \)[/tex]. This indicates a horizontal shift to the left by 3 units.
2. Vertical Scaling and Reflection:
- The function has a multiplier of [tex]\(-\frac{1}{2}\)[/tex] outside the square root.
- The [tex]\(-1/2\)[/tex] factor means we scale the square root function by [tex]\(-1/2\)[/tex], which includes two transformations:
- Reflection: The negative sign reflects the function across the x-axis.
- Vertical Compression: The 1/2 factor compresses the function vertically by half.
3. Vertical Shift Upward:
- There is an additional [tex]\( +2 \)[/tex] outside the square root, which shifts the graph up by 2 units.
To summarize the steps:
- Start from the basic function [tex]\( y = \sqrt{x} \)[/tex].
- Shift it left by 3 to get [tex]\( y = \sqrt{x+3} \)[/tex].
- Reflect and compress vertically by 1/2: [tex]\( y = -\frac{1}{2} \sqrt{x+3} \)[/tex].
- Shift the graph up by 2 units to get the final function: [tex]\( y = -\frac{1}{2} \sqrt{x+3} + 2 \)[/tex].
### Key Points:
To graph this function, it's helpful to calculate and plot specific points by substituting values for [tex]\( x \)[/tex] and solving for [tex]\( y \)[/tex].
1. Determine the domain:
- The expression under the square root (inside √) must be non-negative.
- [tex]\( x + 3 \geq 0 \)[/tex]
- [tex]\( x \geq -3 \)[/tex]
- Thus, the domain is [tex]\( x \geq -3 \)[/tex].
2. Choose key points:
- When [tex]\( x = -3 \)[/tex]:
[tex]\[ y = -\frac{1}{2} \sqrt{-3 + 3} + 2 = -\frac{1}{2} \sqrt{0} + 2 = 2 \][/tex]
- When [tex]\( x = 1 \)[/tex] (a simple value choice):
[tex]\[ y = -\frac{1}{2} \sqrt{1 + 3} + 2 = -\frac{1}{2} \sqrt{4} + 2 = -\frac{1}{2} \cdot 2 + 2 = -1 + 2 = 1 \][/tex]
3. Plot the points and shape:
- Point (-3, 2)
- Point (1, 1)
- Reflecting the downward trend caused by the negative coefficient.
### Graph:
1. Start from point [tex]\((-3, 2)\)[/tex].
2. Draw the curve reflecting the pattern and using additional intermediate values if required.
3. The graph starts from [tex]\((-3, 2)\)[/tex] and decreases as [tex]\( x \)[/tex] increases, smoothly resembling a vertically compressed, reflected half-parabola.
To conclude, by plotting several points and smoothly connecting them, we can draw the correct graph of the function [tex]\( y = -\frac{1}{2} \sqrt{x+3} + 2 \)[/tex]. The graph starts at [tex]\((-3, 2)\)[/tex] and stretches to the right, moving downward, flattened one-half compared to [tex]\( y = -\sqrt{x} \)[/tex].
Thanks for using our platform. We're always here to provide accurate and up-to-date answers to all your queries. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.
