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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.
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