Question
Original graph f(x)=3x^2 -6 with points (0,-6),(-2,4),(2,4),(-2,6) and (2,6) the transformation f(x)=3(0.5x)^2-6 with points (0,-6), (-4,6),(4,6)
0.5x; stretch
0.5x; compression
0.5; stretch
0.5; compression
0.5x; stretch
0.5x; compression
0.5; stretch
0.5; compression
Answers
Answer
Identify the value of k
Answered by
GPT-4o mini
To analyze the transformation of the function \( f(x) = 3x^2 - 6 \) into \( g(x) = 3(0.5x)^2 - 6 \), we can break down the transformation step by step.
1. **Original function**:
\[
f(x) = 3x^2 - 6
\]
The expression \( 3x^2 \) indicates a vertical stretch by a factor of 3, and the \(-6\) shifts the graph down 6 units.
2. **Transformed function**:
\[
g(x) = 3(0.5x)^2 - 6
\]
Here, \( (0.5x) \) means you are compressing the graph horizontally by a factor of \( 0.5 \) (because you are multiplying \( x \) by a number less than 1).
3. **Understanding the transformation**:
- The effect of \( 0.5x \) inside the function means that for every x-coordinate in \( f(x) \), \( g(x) \) takes that x-coordinate to half of its original value.
- This compression by \( 0.5 \) horizontally leads to an increase in the spread of the y-values while retaining the same \( y \) values for the original \( f(x) \).
4. **Points**:
The original points provided were:
- \( (0, -6) \)
- \( (-2, 4) \)
- \( (2, 4) \)
- \( (-2, 6) \)
- \( (2, 6) \)
When we apply the transformation to these points, the new points will be the result of substituting \( 0.5x \):
- \( (0, -6) \) remains \( (0, -6) \)
- \( (-2, 4) \) transforms to \( (-4, 4) \)
- \( (2, 4) \) transforms to \( (4, 4) \)
- The points originally at \( y = 6 \) will also move to the new x-coordinates.
Hence, the correct comprehension of the transformation is that it is a **horizontal compression** by a factor of \( 0.5 \).
### Summary:
- "0.5x" represents a **horizontal compression**.
- The factor \( 3 \) in front of \( x^2 \) indicates a vertical stretch.
- The function will keep the same \( y \)-shift of \(-6\) downward.
Therefore, the primary transformation to note here with respect to \( 0.5x \) is **compression**.
1. **Original function**:
\[
f(x) = 3x^2 - 6
\]
The expression \( 3x^2 \) indicates a vertical stretch by a factor of 3, and the \(-6\) shifts the graph down 6 units.
2. **Transformed function**:
\[
g(x) = 3(0.5x)^2 - 6
\]
Here, \( (0.5x) \) means you are compressing the graph horizontally by a factor of \( 0.5 \) (because you are multiplying \( x \) by a number less than 1).
3. **Understanding the transformation**:
- The effect of \( 0.5x \) inside the function means that for every x-coordinate in \( f(x) \), \( g(x) \) takes that x-coordinate to half of its original value.
- This compression by \( 0.5 \) horizontally leads to an increase in the spread of the y-values while retaining the same \( y \) values for the original \( f(x) \).
4. **Points**:
The original points provided were:
- \( (0, -6) \)
- \( (-2, 4) \)
- \( (2, 4) \)
- \( (-2, 6) \)
- \( (2, 6) \)
When we apply the transformation to these points, the new points will be the result of substituting \( 0.5x \):
- \( (0, -6) \) remains \( (0, -6) \)
- \( (-2, 4) \) transforms to \( (-4, 4) \)
- \( (2, 4) \) transforms to \( (4, 4) \)
- The points originally at \( y = 6 \) will also move to the new x-coordinates.
Hence, the correct comprehension of the transformation is that it is a **horizontal compression** by a factor of \( 0.5 \).
### Summary:
- "0.5x" represents a **horizontal compression**.
- The factor \( 3 \) in front of \( x^2 \) indicates a vertical stretch.
- The function will keep the same \( y \)-shift of \(-6\) downward.
Therefore, the primary transformation to note here with respect to \( 0.5x \) is **compression**.
Answered by
GPT-4o mini
To determine the value of \( k \) in the context of your functions, it appears you may be referring to a transformation of the function \( f(x) = 3x^2 - 6 \) to the form \( g(x) = 3(kx)^2 - 6 \), particularly given the function you are working with, which involves \( 0.5x \).
From the transformation you provided:
\[
g(x) = 3(0.5x)^2 - 6
\]
We can express \( 0.5 \) as \( k \). Therefore, in this case, we can identify \( k = 0.5 \).
If you have a different context in which \( k \) appears or are looking for a different interpretation of what \( k \) represents, please provide further details!
From the transformation you provided:
\[
g(x) = 3(0.5x)^2 - 6
\]
We can express \( 0.5 \) as \( k \). Therefore, in this case, we can identify \( k = 0.5 \).
If you have a different context in which \( k \) appears or are looking for a different interpretation of what \( k \) represents, please provide further details!
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