Westonci.ca offers fast, accurate answers to your questions. Join our community and get the insights you need now. Experience the convenience of getting accurate answers to your questions from a dedicated community of professionals. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
To graph the function [tex]\( y = -\frac{1}{2} \sqrt{x+3} + 2 \)[/tex], we can follow these detailed steps to understand how the graph is formed and what key points and transformations to consider:
### 1. Understanding the Basic Function
Let's start by understanding the standard square root function:
[tex]\[ f(x) = \sqrt{x} \][/tex]
### 2. Transforming the Square Root Function
The given function [tex]\( y = -\frac{1}{2} \sqrt{x+3} + 2 \)[/tex] involves several transformations:
- The expression [tex]\( x+3 \)[/tex] inside the square root indicates a horizontal shift.
- The multiplication by [tex]\( -\frac{1}{2} \)[/tex] affects the vertical stretch and reflection.
- The addition of [tex]\( +2 \)[/tex] at the end shifts the graph vertically.
### 3. Horizontal Shift
The term [tex]\( x+3 \)[/tex] shifts the basic square root function [tex]\( \sqrt{x} \)[/tex] to the left by 3 units.
### 4. Vertical Stretch and Reflection
Multiplying by [tex]\( -\frac{1}{2} \)[/tex] involves:
- A reflection across the x-axis because of the negative sign.
- A vertical compression by a factor of [tex]\(\frac{1}{2} \)[/tex].
### 5. Vertical Shift
Adding 2 to the function moves the entire graph up by 2 units.
### 6. Generating Key Points
We will generate some key points from the transformed function to plot the graph accurately.
Given:
#### X Values:
[tex]\[ \{-3, -2.87, -2.74, -2.61, [-3 \text{ to } 10 \text{ in small intervals]}, 10\} \][/tex]
#### Corresponding Y Values:
[tex]\[ \{2, 1.82, 1.74, 1.69, ..., 0.19\} \][/tex]
### Key Points:
1. When [tex]\( x = -3 \)[/tex]:
[tex]\[ y = -\frac{1}{2} \sqrt{-3+3} + 2 = -\frac{1}{2}\cdot 0 + 2 = 2 \][/tex]
[tex]\((x, y) = (-3, 2)\)[/tex]
2. When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -\frac{1}{2} \sqrt{0+3} + 2 = -\frac{1}{2} \sqrt{3} + 2 \approx 1.13 \][/tex]
[tex]\((x, y) = (0, 1.13)\)[/tex]
3. When [tex]\( x = 1 \)[/tex]:
[tex]\[ y = -\frac{1}{2} \sqrt{1+3} + 2 = -\frac{1}{2} \sqrt{4} + 2 = -1 + 2 = 1 \][/tex]
[tex]\((x, y) = (1, 1)\)[/tex]
4. When [tex]\( x = 4 \)[/tex]:
[tex]\[ y = -\frac{1}{2} \sqrt{4+3} + 2 = -\frac{1}{2} \sqrt{7} + 2 \approx 0.68 \][/tex]
[tex]\((x, y) = (4, 0.68)\)[/tex]
For additional detail, you can evaluate several more points within the range [tex]\([-3, 10]\)[/tex] to generate a smoother plot.
### 7. Plotting the Graph
By plotting these points and observing the function’s behavior:
1. Begin by plotting the point [tex]\((-3, 2)\)[/tex].
2. Plot additional points along the calculated x values and their corresponding y values.
3. Connect the points smoothly, considering the function’s gradual decline due to the negative coefficient and vertical compression.
### Summary
The graph of the function [tex]\( y = -\frac{1}{2} \sqrt{x+3} + 2 \)[/tex] will start at the point [tex]\((-3, 2)\)[/tex] and curve downwards, gradually approaching closer to a horizontal line as [tex]\( x \)[/tex] increases. The function is defined for [tex]\( x \geq -3 \)[/tex]. The changes in the y values will reflect the transformations we've discussed, showing the impact of the vertical compression, reflection, and shifts.
### 1. Understanding the Basic Function
Let's start by understanding the standard square root function:
[tex]\[ f(x) = \sqrt{x} \][/tex]
### 2. Transforming the Square Root Function
The given function [tex]\( y = -\frac{1}{2} \sqrt{x+3} + 2 \)[/tex] involves several transformations:
- The expression [tex]\( x+3 \)[/tex] inside the square root indicates a horizontal shift.
- The multiplication by [tex]\( -\frac{1}{2} \)[/tex] affects the vertical stretch and reflection.
- The addition of [tex]\( +2 \)[/tex] at the end shifts the graph vertically.
### 3. Horizontal Shift
The term [tex]\( x+3 \)[/tex] shifts the basic square root function [tex]\( \sqrt{x} \)[/tex] to the left by 3 units.
### 4. Vertical Stretch and Reflection
Multiplying by [tex]\( -\frac{1}{2} \)[/tex] involves:
- A reflection across the x-axis because of the negative sign.
- A vertical compression by a factor of [tex]\(\frac{1}{2} \)[/tex].
### 5. Vertical Shift
Adding 2 to the function moves the entire graph up by 2 units.
### 6. Generating Key Points
We will generate some key points from the transformed function to plot the graph accurately.
Given:
#### X Values:
[tex]\[ \{-3, -2.87, -2.74, -2.61, [-3 \text{ to } 10 \text{ in small intervals]}, 10\} \][/tex]
#### Corresponding Y Values:
[tex]\[ \{2, 1.82, 1.74, 1.69, ..., 0.19\} \][/tex]
### Key Points:
1. When [tex]\( x = -3 \)[/tex]:
[tex]\[ y = -\frac{1}{2} \sqrt{-3+3} + 2 = -\frac{1}{2}\cdot 0 + 2 = 2 \][/tex]
[tex]\((x, y) = (-3, 2)\)[/tex]
2. When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -\frac{1}{2} \sqrt{0+3} + 2 = -\frac{1}{2} \sqrt{3} + 2 \approx 1.13 \][/tex]
[tex]\((x, y) = (0, 1.13)\)[/tex]
3. When [tex]\( x = 1 \)[/tex]:
[tex]\[ y = -\frac{1}{2} \sqrt{1+3} + 2 = -\frac{1}{2} \sqrt{4} + 2 = -1 + 2 = 1 \][/tex]
[tex]\((x, y) = (1, 1)\)[/tex]
4. When [tex]\( x = 4 \)[/tex]:
[tex]\[ y = -\frac{1}{2} \sqrt{4+3} + 2 = -\frac{1}{2} \sqrt{7} + 2 \approx 0.68 \][/tex]
[tex]\((x, y) = (4, 0.68)\)[/tex]
For additional detail, you can evaluate several more points within the range [tex]\([-3, 10]\)[/tex] to generate a smoother plot.
### 7. Plotting the Graph
By plotting these points and observing the function’s behavior:
1. Begin by plotting the point [tex]\((-3, 2)\)[/tex].
2. Plot additional points along the calculated x values and their corresponding y values.
3. Connect the points smoothly, considering the function’s gradual decline due to the negative coefficient and vertical compression.
### Summary
The graph of the function [tex]\( y = -\frac{1}{2} \sqrt{x+3} + 2 \)[/tex] will start at the point [tex]\((-3, 2)\)[/tex] and curve downwards, gradually approaching closer to a horizontal line as [tex]\( x \)[/tex] increases. The function is defined for [tex]\( x \geq -3 \)[/tex]. The changes in the y values will reflect the transformations we've discussed, showing the impact of the vertical compression, reflection, and shifts.
Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
