Question
When transforming f(x)=x^3 to f(x)=−(2/3x+3)^3 +2, what transformations should be performed and in what order?(1 point)
Responses
horizontal stretch, translated horizontally to the right, and reflected across the x-axis, then translated vertically up
horizontal stretch, translated horizontally to the right, and reflected across the x -axis, then translated vertically up
horizontal stretch, translated horizontally to the left, and reflected across the y-axis, then translated vertically up
horizontal stretch, translated horizontally to the left, and reflected across the y -axis, then translated vertically up
translated horizontally to the left, horizontal stretch, and reflected across the x-axis, then translated vertically up
translated horizontally to the left, horizontal stretch, and reflected across the x -axis, then translated vertically up
horizontal stretch, translated horizontally to the left, and reflected across the x-axis, then translated vertically up
Responses
horizontal stretch, translated horizontally to the right, and reflected across the x-axis, then translated vertically up
horizontal stretch, translated horizontally to the right, and reflected across the x -axis, then translated vertically up
horizontal stretch, translated horizontally to the left, and reflected across the y-axis, then translated vertically up
horizontal stretch, translated horizontally to the left, and reflected across the y -axis, then translated vertically up
translated horizontally to the left, horizontal stretch, and reflected across the x-axis, then translated vertically up
translated horizontally to the left, horizontal stretch, and reflected across the x -axis, then translated vertically up
horizontal stretch, translated horizontally to the left, and reflected across the x-axis, then translated vertically up
Answers
GPT-4o mini
To transform the function \( f(x) = x^3 \) to \( f(x) = -\left(\frac{2}{3}x + 3\right)^3 + 2 \), we can analyze the transformations step-by-step.
1. **Start with horizontal stretch:** The term \( \frac{2}{3}x \) indicates a horizontal stretch by a factor of \( \frac{3}{2} \).
2. **Translate horizontally:** The term \( +3 \) inside the parenthesis indicates a translation horizontally to the left by 3 units.
3. **Reflect across the x-axis:** The negative sign outside the cube reflects the graph across the x-axis.
4. **Translate vertically:** Finally, the \( +2 \) translates the graph vertically up by 2 units.
Putting this together, the transformations are:
1. Horizontal stretch (by a factor of \( \frac{3}{2} \))
2. Translate horizontally to the left (by 3 units)
3. Reflect across the x-axis
4. Translate vertically up (by 2 units)
Therefore, the correct sequence of transformations is:
**horizontal stretch, translated horizontally to the left, and reflected across the x-axis, then translated vertically up.**
1. **Start with horizontal stretch:** The term \( \frac{2}{3}x \) indicates a horizontal stretch by a factor of \( \frac{3}{2} \).
2. **Translate horizontally:** The term \( +3 \) inside the parenthesis indicates a translation horizontally to the left by 3 units.
3. **Reflect across the x-axis:** The negative sign outside the cube reflects the graph across the x-axis.
4. **Translate vertically:** Finally, the \( +2 \) translates the graph vertically up by 2 units.
Putting this together, the transformations are:
1. Horizontal stretch (by a factor of \( \frac{3}{2} \))
2. Translate horizontally to the left (by 3 units)
3. Reflect across the x-axis
4. Translate vertically up (by 2 units)
Therefore, the correct sequence of transformations is:
**horizontal stretch, translated horizontally to the left, and reflected across the x-axis, then translated vertically up.**