Question
Describe the transformations that bring g(x)−4x+1−−−−−√−3
to h(x)=−234x+1−−−−−√−3
.(1 point)
Responses
reflection over the line x=−3
and a vertical compression by a factor of 23
reflection over the line x is equal to negative 3 and a vertical compression by a factor of 2 thirds
reflection over the line y-axis and a vertical stretch by a factor of 32
reflection over the line y -axis and a vertical stretch by a factor of 3 halves
reflection over the line x=−3
and a vertical compression by a factor of 32
reflection over the line x is equal to negative 3 and a vertical compression by a factor of 3 halves
reflection over the line y=−3
and a vertical compression by a factor of 23
reflection over the line y is equal to negative 3 and a vertical compression by a factor of 2 thirds
to h(x)=−234x+1−−−−−√−3
.(1 point)
Responses
reflection over the line x=−3
and a vertical compression by a factor of 23
reflection over the line x is equal to negative 3 and a vertical compression by a factor of 2 thirds
reflection over the line y-axis and a vertical stretch by a factor of 32
reflection over the line y -axis and a vertical stretch by a factor of 3 halves
reflection over the line x=−3
and a vertical compression by a factor of 32
reflection over the line x is equal to negative 3 and a vertical compression by a factor of 3 halves
reflection over the line y=−3
and a vertical compression by a factor of 23
reflection over the line y is equal to negative 3 and a vertical compression by a factor of 2 thirds
Answers
Answered by
GPT-4o mini
To transform the function \( g(x) = \sqrt{-4x + 1} - 3 \) into \( h(x) = -\frac{2}{3} \sqrt{x + 1} - 3 \), we need to analyze the changes systematically.
1. **Horizontal Translation**: Notice that \( g(x) \) has the new variable inside the square root. The transformation from \( -4x + 1 \) to \( x + 1 \) can be understood through the substitution affecting the input.
- The change from \( -4x \) to \( x \) can be seen as a reflection and a horizontal stretch. Specifically, dividing by \(-4\) (which reflects across the y-axis) allows for the transformation.
2. **Vertical Transformation**: The square root has a negative sign in front of it in \( h(x) \). This means we need to reflect \( g(x) \) over the horizontal axis (y-axis).
3. **Vertical Compression**: The coefficient of \(-\frac{2}{3}\) means we have a vertical compression by a factor of \( \frac{2}{3} \).
Putting it all together, the transformations needed to convert \( g(x) \) to \( h(x) \) include reflecting over the y-axis and a vertical compression by a factor of \( \frac{2}{3} \).
Thus, the correct choice from your options would be:
**Reflection over the line \( y = -3 \) and a vertical compression by a factor of \( \frac{2}{3} \)**.
(Note: Make sure to verify if any additional details are required in options that reflect the line \( x = -3 \) or more distinctly detail the transformations involved.)
1. **Horizontal Translation**: Notice that \( g(x) \) has the new variable inside the square root. The transformation from \( -4x + 1 \) to \( x + 1 \) can be understood through the substitution affecting the input.
- The change from \( -4x \) to \( x \) can be seen as a reflection and a horizontal stretch. Specifically, dividing by \(-4\) (which reflects across the y-axis) allows for the transformation.
2. **Vertical Transformation**: The square root has a negative sign in front of it in \( h(x) \). This means we need to reflect \( g(x) \) over the horizontal axis (y-axis).
3. **Vertical Compression**: The coefficient of \(-\frac{2}{3}\) means we have a vertical compression by a factor of \( \frac{2}{3} \).
Putting it all together, the transformations needed to convert \( g(x) \) to \( h(x) \) include reflecting over the y-axis and a vertical compression by a factor of \( \frac{2}{3} \).
Thus, the correct choice from your options would be:
**Reflection over the line \( y = -3 \) and a vertical compression by a factor of \( \frac{2}{3} \)**.
(Note: Make sure to verify if any additional details are required in options that reflect the line \( x = -3 \) or more distinctly detail the transformations involved.)