describe the transformations that happen to the graph of f(x)=sin x to obtain the graph of f(x)=-2sin(x-3)+7

Question

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

Answer 1

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.

Answer 2

Are you sure

Answer 3

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!