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Sagot :
Sure, let's break down this question step by step to identify the correct transformations applied to the graph of [tex]\( f(x) = x^2 \)[/tex] to obtain [tex]\( g(x) = -3x^2 - 36x \)[/tex].
First, let's understand what each term in [tex]\( g(x) \)[/tex] does to the graph of [tex]\( f(x) \)[/tex]:
1. Transformation by the coefficient in front of [tex]\( x^2 \)[/tex]:
- The term [tex]\( -3x^2 \)[/tex] indicates a reflection across the x-axis and a vertical stretch by a factor of 3. This makes the graph narrower compared to the graph of [tex]\( f(x) = x^2 \)[/tex].
2. Horizontal shift and stretch/compression:
- The term [tex]\( -36x \)[/tex] affects the position of the vertex of the parabola. To find the vertex form of [tex]\( -3x^2 - 36x \)[/tex], you need to complete the square:
[tex]\[ g(x) = -3x^2 - 36x = -3(x^2 + 12x) \][/tex]
To complete the square, you take half of the coefficient of [tex]\( x \)[/tex] (which is 12), square it to get 36, and then add and subtract 36 inside the parentheses:
[tex]\[ -3(x^2 + 12x + 36 - 36) = -3((x + 6)^2 - 36) = -3(x + 6)^2 + 108 \][/tex]
- This shows the parabola is shifted horizontally to the left by 6 units and vertically up by 108 units due to the constant term.
Next, we match the transformations described in the options to those identified:
1. The graph of [tex]\( f(x) = x^2 \)[/tex] is made narrower:
- Yes, the coefficient [tex]\( -3 \)[/tex] makes the graph narrower.
2. The graph of [tex]\( f(x) = x^2 \)[/tex] is shifted right 6 units:
- The completed square form shows a shift left by 6 units, not right.
3. The graph of [tex]\( f(x) = x^2 \)[/tex] is shifted down 48 units:
- According to our completed square form, the graph is shifted up by 108 units, not down by 48.
4. The graph of [tex]\( f(x) = x^2 \)[/tex] is reflected over the y-axis:
- There is no indication of reflection over the y-axis; the reflection is over the x-axis due to the negative coefficient.
From these explanations:
- Transformation 1 (The graph is made narrower) is true.
- Transformation 4 (Reflection over the y-axis) is incorrect, but the equation does involve reflection over the x-axis.
Thus, the transformations applied include 'made narrower' and reflection over the x-axis.
The applied transformations that are correct according to the analysis are:
- Transformation: The graph of [tex]\( f(x)=x^2 \)[/tex] is made narrower.
- Transformation: Reflection over the x-axis.
The transformations indexed at 1, [tex]\( 4 \)[/tex] in your list. Therefore, the provided solution indicates:
```plaintext
[1, 4]
```
First, let's understand what each term in [tex]\( g(x) \)[/tex] does to the graph of [tex]\( f(x) \)[/tex]:
1. Transformation by the coefficient in front of [tex]\( x^2 \)[/tex]:
- The term [tex]\( -3x^2 \)[/tex] indicates a reflection across the x-axis and a vertical stretch by a factor of 3. This makes the graph narrower compared to the graph of [tex]\( f(x) = x^2 \)[/tex].
2. Horizontal shift and stretch/compression:
- The term [tex]\( -36x \)[/tex] affects the position of the vertex of the parabola. To find the vertex form of [tex]\( -3x^2 - 36x \)[/tex], you need to complete the square:
[tex]\[ g(x) = -3x^2 - 36x = -3(x^2 + 12x) \][/tex]
To complete the square, you take half of the coefficient of [tex]\( x \)[/tex] (which is 12), square it to get 36, and then add and subtract 36 inside the parentheses:
[tex]\[ -3(x^2 + 12x + 36 - 36) = -3((x + 6)^2 - 36) = -3(x + 6)^2 + 108 \][/tex]
- This shows the parabola is shifted horizontally to the left by 6 units and vertically up by 108 units due to the constant term.
Next, we match the transformations described in the options to those identified:
1. The graph of [tex]\( f(x) = x^2 \)[/tex] is made narrower:
- Yes, the coefficient [tex]\( -3 \)[/tex] makes the graph narrower.
2. The graph of [tex]\( f(x) = x^2 \)[/tex] is shifted right 6 units:
- The completed square form shows a shift left by 6 units, not right.
3. The graph of [tex]\( f(x) = x^2 \)[/tex] is shifted down 48 units:
- According to our completed square form, the graph is shifted up by 108 units, not down by 48.
4. The graph of [tex]\( f(x) = x^2 \)[/tex] is reflected over the y-axis:
- There is no indication of reflection over the y-axis; the reflection is over the x-axis due to the negative coefficient.
From these explanations:
- Transformation 1 (The graph is made narrower) is true.
- Transformation 4 (Reflection over the y-axis) is incorrect, but the equation does involve reflection over the x-axis.
Thus, the transformations applied include 'made narrower' and reflection over the x-axis.
The applied transformations that are correct according to the analysis are:
- Transformation: The graph of [tex]\( f(x)=x^2 \)[/tex] is made narrower.
- Transformation: Reflection over the x-axis.
The transformations indexed at 1, [tex]\( 4 \)[/tex] in your list. Therefore, the provided solution indicates:
```plaintext
[1, 4]
```
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