Question
describe the transformations that happen to the graph of f(x)=sin x to obtain the graph of f(x)=-2sin(x-3)+7
A. Vertical stretch of 2 reflect over the y axis right 3 units and up 7 units
B. Vertical stretch of 2 reflect over the y axis Left 3 units and up 7 units
C. Vertical stretch of 2 reflect over the y axis Right 3 units and down 7 units
D. Vertical stretch of 2 reflect over the y axis Right 3 units and up 7 units
A. Vertical stretch of 2 reflect over the y axis right 3 units and up 7 units
B. Vertical stretch of 2 reflect over the y axis Left 3 units and up 7 units
C. Vertical stretch of 2 reflect over the y axis Right 3 units and down 7 units
D. Vertical stretch of 2 reflect over the y axis Right 3 units and up 7 units
Answers
Answered by
GPT-4o mini
To obtain the graph of \( f(x) = -2\sin(x - 3) + 7 \) from the graph of \( f(x) = \sin x \), we can analyze the transformations step by step:
1. **Horizontal Shift**: The term \( (x - 3) \) indicates a shift to the right by 3 units. This transformation moves the entire graph to the right along the x-axis.
2. **Vertical Stretch and Reflection**: The coefficient \(-2\) indicates two transformations:
- The negative sign indicates a reflection over the x-axis (not the y-axis).
- The number 2 indicates a vertical stretch by a factor of 2.
3. **Vertical Shift**: The \( +7 \) indicates a shift upward by 7 units.
Putting it all together, the transformations are:
- Shift right by 3 units
- Reflect over the x-axis
- Stretch vertically by a factor of 2
- Shift up by 7 units
None of the options mention the reflection over the x-axis, which is crucial. However, focusing only on the shifts and vertical stretching factors provided in the options, the correct option according to the shifts and stretch is:
**D. Vertical stretch of 2 reflect over the y axis Right 3 units and up 7 units.**
However, it's important to note that "reflect over the y-axis" is incorrect; it should be "reflect over the x-axis." Despite this error, based on the general structure of the answer options, D is the closest choice if we ignore the reflection detail.
1. **Horizontal Shift**: The term \( (x - 3) \) indicates a shift to the right by 3 units. This transformation moves the entire graph to the right along the x-axis.
2. **Vertical Stretch and Reflection**: The coefficient \(-2\) indicates two transformations:
- The negative sign indicates a reflection over the x-axis (not the y-axis).
- The number 2 indicates a vertical stretch by a factor of 2.
3. **Vertical Shift**: The \( +7 \) indicates a shift upward by 7 units.
Putting it all together, the transformations are:
- Shift right by 3 units
- Reflect over the x-axis
- Stretch vertically by a factor of 2
- Shift up by 7 units
None of the options mention the reflection over the x-axis, which is crucial. However, focusing only on the shifts and vertical stretching factors provided in the options, the correct option according to the shifts and stretch is:
**D. Vertical stretch of 2 reflect over the y axis Right 3 units and up 7 units.**
However, it's important to note that "reflect over the y-axis" is incorrect; it should be "reflect over the x-axis." Despite this error, based on the general structure of the answer options, D is the closest choice if we ignore the reflection detail.
Answer
Are you sure
Answered by
GPT-4o mini
Let's clarify the transformations step by step once again to ensure accuracy:
We start with the function \( f(x) = \sin x \), and we want to transform it into \( f(x) = -2\sin(x - 3) + 7 \).
1. **Horizontal Shift**:
- The term \( (x - 3) \) indicates that the graph is shifted **to the right by 3 units**.
2. **Reflection**:
- The negative sign in front of the \( 2 \) (i.e., \(-2\)) indicates a **reflection over the x-axis**.
3. **Vertical Stretch**:
- The coefficient \( 2 \) indicates a **vertical stretch by a factor of 2**. This means the amplitude of the sine function, which normally ranges from \(-1\) to \(1\), will now range from \(-2\) to \(2\) before the vertical shift.
4. **Vertical Shift**:
- The \( +7 \) at the end indicates that the entire graph is shifted **up by 7 units**.
Now, let's summarize the transformations correctly:
- Shift **right** by **3 units** (due to \( x - 3 \))
- **Reflect over the x-axis** (due to the negative sign)
- **Vertical stretch** by a factor of **2**
- Shift **up** by **7 units**
Now let's review the answer choices:
A. Vertical stretch of 2 reflect over the y-axis right 3 units and up 7 units
B. Vertical stretch of 2 reflect over the y-axis left 3 units and up 7 units
C. Vertical stretch of 2 reflect over the y-axis right 3 units and down 7 units
D. Vertical stretch of 2 reflect over the y-axis right 3 units and up 7 units
Given this breakdown, none of the choices are entirely accurate. They incorrectly state a reflection over the **y-axis** when it should be a reflection over the **x-axis**.
However, if we disregard this error in the options and focus on the transformations:
- The transformation regarding the "right 3 units" and "up 7 units" is correct in D, but the reflection should state **reflection over the x-axis** instead.
So, if you’re looking for the option that accurately describes the shifts but has a minor error in the reflection, it would still be close to **D**. However, please be aware of the critical note regarding the axis of reflection. Thank you for your patience, and I appreciate the opportunity to clarify!
We start with the function \( f(x) = \sin x \), and we want to transform it into \( f(x) = -2\sin(x - 3) + 7 \).
1. **Horizontal Shift**:
- The term \( (x - 3) \) indicates that the graph is shifted **to the right by 3 units**.
2. **Reflection**:
- The negative sign in front of the \( 2 \) (i.e., \(-2\)) indicates a **reflection over the x-axis**.
3. **Vertical Stretch**:
- The coefficient \( 2 \) indicates a **vertical stretch by a factor of 2**. This means the amplitude of the sine function, which normally ranges from \(-1\) to \(1\), will now range from \(-2\) to \(2\) before the vertical shift.
4. **Vertical Shift**:
- The \( +7 \) at the end indicates that the entire graph is shifted **up by 7 units**.
Now, let's summarize the transformations correctly:
- Shift **right** by **3 units** (due to \( x - 3 \))
- **Reflect over the x-axis** (due to the negative sign)
- **Vertical stretch** by a factor of **2**
- Shift **up** by **7 units**
Now let's review the answer choices:
A. Vertical stretch of 2 reflect over the y-axis right 3 units and up 7 units
B. Vertical stretch of 2 reflect over the y-axis left 3 units and up 7 units
C. Vertical stretch of 2 reflect over the y-axis right 3 units and down 7 units
D. Vertical stretch of 2 reflect over the y-axis right 3 units and up 7 units
Given this breakdown, none of the choices are entirely accurate. They incorrectly state a reflection over the **y-axis** when it should be a reflection over the **x-axis**.
However, if we disregard this error in the options and focus on the transformations:
- The transformation regarding the "right 3 units" and "up 7 units" is correct in D, but the reflection should state **reflection over the x-axis** instead.
So, if you’re looking for the option that accurately describes the shifts but has a minor error in the reflection, it would still be close to **D**. However, please be aware of the critical note regarding the axis of reflection. Thank you for your patience, and I appreciate the opportunity to clarify!