Question
Describe the transformation of the parent function h(x)=3x
to turn into k(x)=2⋅3−12x−5
(1 point)
Responses
Reflection across the y-axis, horizontal shrink by a factor of 2, vertical stretch by a factor of 2, vertical shift upwards by 5 units.
Reflection across the y-axis, horizontal shrink by a factor of 2, vertical stretch by a factor of 2, vertical shift upwards by 5 units.
Reflection across the x-axis, horizontal stretch by a factor of 2, vertical shrink by a factor of 2, vertical shift upwards by 5 units.
Reflection across the x-axis, horizontal stretch by a factor of 2, vertical shrink by a factor of 2, vertical shift upwards by 5 units.
Reflection across the y-axis, horizontal stretch by a factor of 2, vertical stretch by a factor of 2, vertical shift downwards by 5 units.
Reflection across the y-axis, horizontal stretch by a factor of 2, vertical stretch by a factor of 2, vertical shift downwards by 5 units.
Reflection across the y-axis, horizontal shrink by a factor of 2, vertical stretch by a factor of 2, vertical shift downwards by 5 units.
to turn into k(x)=2⋅3−12x−5
(1 point)
Responses
Reflection across the y-axis, horizontal shrink by a factor of 2, vertical stretch by a factor of 2, vertical shift upwards by 5 units.
Reflection across the y-axis, horizontal shrink by a factor of 2, vertical stretch by a factor of 2, vertical shift upwards by 5 units.
Reflection across the x-axis, horizontal stretch by a factor of 2, vertical shrink by a factor of 2, vertical shift upwards by 5 units.
Reflection across the x-axis, horizontal stretch by a factor of 2, vertical shrink by a factor of 2, vertical shift upwards by 5 units.
Reflection across the y-axis, horizontal stretch by a factor of 2, vertical stretch by a factor of 2, vertical shift downwards by 5 units.
Reflection across the y-axis, horizontal stretch by a factor of 2, vertical stretch by a factor of 2, vertical shift downwards by 5 units.
Reflection across the y-axis, horizontal shrink by a factor of 2, vertical stretch by a factor of 2, vertical shift downwards by 5 units.
Answers
Answered by
GPT-4o mini
To analyze the transformation from the parent function \( h(x) = 3^x \) to \( k(x) = 2 \cdot 3^{-12x} - 5 \), we can break down the transformations step by step.
1. **Starting with the Parent Function**:
The parent function is \( h(x) = 3^x \).
2. **Horizontal Reflection**:
The \( 3^{-12x} \) indicates a reflection across the y-axis because of the negative exponent. This changes the function from \( 3^x \) to \( 3^{-12x} \).
3. **Horizontal Compression**:
The factor of \( 12 \) in \( 3^{-12x} \) represents a horizontal shrink by a factor of \( \frac{1}{12} \) (or compression).
4. **Vertical Stretch**:
The multiplication by \( 2 \) indicates a vertical stretch by a factor of \( 2 \).
5. **Vertical Shift**:
The \( -5 \) indicates a vertical shift downwards by \( 5 \) units.
To summarize the transformations:
- Reflection across the y-axis (due to the negative exponent).
- Horizontal shrink by a factor of \( \frac{1}{12} \) (since \( -12x \)).
- Vertical stretch by a factor of \( 2 \).
- Vertical shift downwards by \( 5 \) units.
### Conclusion:
Considering all the transformations, the closest match from the provided options would be:
"**Reflection across the y-axis, horizontal shrink by a factor of 12, vertical stretch by a factor of 2, vertical shift downwards by 5 units.**"
However, since this exact wording does not precisely exist in the provided choices (correct factor not matching), the potential answers require careful consideration of each option's specifics. Hence the answer is not directly present as stated but aligns conceptually with the last option regarding transformations but with greater specificity in context.
If we strictly go by the option structures presented, it can be said that none of the provided responses are a full match upon careful observation but engage largely in horizontal shift and reflections applied to modified factors therein.
1. **Starting with the Parent Function**:
The parent function is \( h(x) = 3^x \).
2. **Horizontal Reflection**:
The \( 3^{-12x} \) indicates a reflection across the y-axis because of the negative exponent. This changes the function from \( 3^x \) to \( 3^{-12x} \).
3. **Horizontal Compression**:
The factor of \( 12 \) in \( 3^{-12x} \) represents a horizontal shrink by a factor of \( \frac{1}{12} \) (or compression).
4. **Vertical Stretch**:
The multiplication by \( 2 \) indicates a vertical stretch by a factor of \( 2 \).
5. **Vertical Shift**:
The \( -5 \) indicates a vertical shift downwards by \( 5 \) units.
To summarize the transformations:
- Reflection across the y-axis (due to the negative exponent).
- Horizontal shrink by a factor of \( \frac{1}{12} \) (since \( -12x \)).
- Vertical stretch by a factor of \( 2 \).
- Vertical shift downwards by \( 5 \) units.
### Conclusion:
Considering all the transformations, the closest match from the provided options would be:
"**Reflection across the y-axis, horizontal shrink by a factor of 12, vertical stretch by a factor of 2, vertical shift downwards by 5 units.**"
However, since this exact wording does not precisely exist in the provided choices (correct factor not matching), the potential answers require careful consideration of each option's specifics. Hence the answer is not directly present as stated but aligns conceptually with the last option regarding transformations but with greater specificity in context.
If we strictly go by the option structures presented, it can be said that none of the provided responses are a full match upon careful observation but engage largely in horizontal shift and reflections applied to modified factors therein.