Answer :
To determine which function [tex]\( g(x) \)[/tex] is the correct transformation of the given quadratic parent function [tex]\( f(x) = x^2 \)[/tex], we need to evaluate the potential transformations and their impacts.
1. Understanding the Parent Function:
The parent function [tex]\( f(x) = x^2 \)[/tex] is a quadratic function which produces a parabola opening upwards with its vertex at the origin (0, 0).
2. Types of Transformations:
We can apply vertical stretches, compressions, and reflections to modify this parent function. These transformations typically involve multiplying the function [tex]\( f(x) \)[/tex] by a constant factor [tex]\( a \)[/tex].
- If [tex]\( a > 1 \)[/tex], the function [tex]\( f(x) \)[/tex] undergoes a vertical stretch.
- If [tex]\( 0 < a < 1 \)[/tex], the function [tex]\( f(x) \)[/tex] undergoes a vertical compression.
- If [tex]\( a < 0 \)[/tex], the function [tex]\( f(x) \)[/tex] is reflected over the x-axis.
3. Analyzing the Given Options:
- Option A: [tex]\( g(x) = 3x^2 \)[/tex]
- This suggests a vertical stretch by a factor of 3.
- Option B: [tex]\( g(x) = -\frac{1}{3}x^2 \)[/tex]
- This suggests a vertical compression by a factor of [tex]\(\frac{1}{3}\)[/tex] and a reflection over the x-axis.
- Option C: [tex]\( g(x) = \frac{1}{3}x^2 \)[/tex]
- This suggests a vertical compression by a factor of [tex]\(\frac{1}{3}\)[/tex].
- Option D: [tex]\( g(x) = -3x^2 \)[/tex]
- This suggests a vertical stretch by a factor of 3 and a reflection over the x-axis.
4. Evaluating Transformations Numerically:
To make our analysis comprehensive, let's examine the value [tex]\( g(1) \)[/tex] using the potential transformations:
- For [tex]\( f(x) = x^2 \)[/tex], [tex]\( f(1) = 1^2 = 1 \)[/tex].
- Option A: [tex]\( g(1) = 3 \cdot (1)^2 = 3 \)[/tex]
- Result: [tex]\( 3 \)[/tex]
- Option B: [tex]\( g(1) = -\frac{1}{3} \cdot (1)^2 = -\frac{1}{3} \)[/tex]
- Result: [tex]\( -0.333333 \)[/tex]
- Option C: [tex]\( g(1) = \frac{1}{3} \cdot (1)^2 = \frac{1}{3} \)[/tex]
- Result: [tex]\( 0.333333 \)[/tex]
- Option D: [tex]\( g(1) = -3 \cdot (1)^2 = -3 \)[/tex]
- Result: [tex]\( -3 \)[/tex]
5. Comparing Results:
When we compare the results:
- Option A gives [tex]\( g(1) = 3 \)[/tex],
- Option B gives [tex]\( g(1) = -0.333333 \)[/tex],
- Option C gives [tex]\( g(1) = 0.333333 \)[/tex],
- Option D gives [tex]\( g(1) = -3 \)[/tex].
Based on the analysis of these transformations and comparing the outcomes, the four transformation options provided (A, B, C, and D) lead to the following correct functions [tex]\( g(x) \)[/tex]:
- A: [tex]\( g(x) = 3x^2 \)[/tex]
- B: [tex]\( g(x) = -\frac{1}{3}x^2 \)[/tex]
- C: [tex]\( g(x) = \frac{1}{3}x^2 \)[/tex]
- D: [tex]\( g(x) = -3x^2 \)[/tex]
Each of these corresponds to one of the described transformations in the question. Thus, all the provided functions (A, B, C, and D) are valid transformations of the parent function [tex]\( f(x) = x^2 \)[/tex] given the correct context.
However, if you are asked to specifically identify which transformations correspond to the given absolute results [tex]\( [3, -0.3333333333333333, 0.3333333333333333, -3] \)[/tex] for [tex]\( g(1) \)[/tex]:
- [tex]\(3\)[/tex] corresponds to [tex]\( g(x) = 3x^2 \)[/tex]
- [tex]\(-0.333333\)[/tex] corresponds to [tex]\( g(x) = -\frac{1}{3}x^2 \)[/tex]
- [tex]\(0.333333\)[/tex] corresponds to [tex]\( g(x) = \frac{1}{3}x^2 \)[/tex]
- [tex]\(-3\)[/tex] corresponds to [tex]\( g(x) = -3x^2 \)[/tex]
Therefore, all the given functions (A, B, C, and D) are transformations of [tex]\( f(x) = x^2 \)[/tex] with valid, corresponding results.
1. Understanding the Parent Function:
The parent function [tex]\( f(x) = x^2 \)[/tex] is a quadratic function which produces a parabola opening upwards with its vertex at the origin (0, 0).
2. Types of Transformations:
We can apply vertical stretches, compressions, and reflections to modify this parent function. These transformations typically involve multiplying the function [tex]\( f(x) \)[/tex] by a constant factor [tex]\( a \)[/tex].
- If [tex]\( a > 1 \)[/tex], the function [tex]\( f(x) \)[/tex] undergoes a vertical stretch.
- If [tex]\( 0 < a < 1 \)[/tex], the function [tex]\( f(x) \)[/tex] undergoes a vertical compression.
- If [tex]\( a < 0 \)[/tex], the function [tex]\( f(x) \)[/tex] is reflected over the x-axis.
3. Analyzing the Given Options:
- Option A: [tex]\( g(x) = 3x^2 \)[/tex]
- This suggests a vertical stretch by a factor of 3.
- Option B: [tex]\( g(x) = -\frac{1}{3}x^2 \)[/tex]
- This suggests a vertical compression by a factor of [tex]\(\frac{1}{3}\)[/tex] and a reflection over the x-axis.
- Option C: [tex]\( g(x) = \frac{1}{3}x^2 \)[/tex]
- This suggests a vertical compression by a factor of [tex]\(\frac{1}{3}\)[/tex].
- Option D: [tex]\( g(x) = -3x^2 \)[/tex]
- This suggests a vertical stretch by a factor of 3 and a reflection over the x-axis.
4. Evaluating Transformations Numerically:
To make our analysis comprehensive, let's examine the value [tex]\( g(1) \)[/tex] using the potential transformations:
- For [tex]\( f(x) = x^2 \)[/tex], [tex]\( f(1) = 1^2 = 1 \)[/tex].
- Option A: [tex]\( g(1) = 3 \cdot (1)^2 = 3 \)[/tex]
- Result: [tex]\( 3 \)[/tex]
- Option B: [tex]\( g(1) = -\frac{1}{3} \cdot (1)^2 = -\frac{1}{3} \)[/tex]
- Result: [tex]\( -0.333333 \)[/tex]
- Option C: [tex]\( g(1) = \frac{1}{3} \cdot (1)^2 = \frac{1}{3} \)[/tex]
- Result: [tex]\( 0.333333 \)[/tex]
- Option D: [tex]\( g(1) = -3 \cdot (1)^2 = -3 \)[/tex]
- Result: [tex]\( -3 \)[/tex]
5. Comparing Results:
When we compare the results:
- Option A gives [tex]\( g(1) = 3 \)[/tex],
- Option B gives [tex]\( g(1) = -0.333333 \)[/tex],
- Option C gives [tex]\( g(1) = 0.333333 \)[/tex],
- Option D gives [tex]\( g(1) = -3 \)[/tex].
Based on the analysis of these transformations and comparing the outcomes, the four transformation options provided (A, B, C, and D) lead to the following correct functions [tex]\( g(x) \)[/tex]:
- A: [tex]\( g(x) = 3x^2 \)[/tex]
- B: [tex]\( g(x) = -\frac{1}{3}x^2 \)[/tex]
- C: [tex]\( g(x) = \frac{1}{3}x^2 \)[/tex]
- D: [tex]\( g(x) = -3x^2 \)[/tex]
Each of these corresponds to one of the described transformations in the question. Thus, all the provided functions (A, B, C, and D) are valid transformations of the parent function [tex]\( f(x) = x^2 \)[/tex] given the correct context.
However, if you are asked to specifically identify which transformations correspond to the given absolute results [tex]\( [3, -0.3333333333333333, 0.3333333333333333, -3] \)[/tex] for [tex]\( g(1) \)[/tex]:
- [tex]\(3\)[/tex] corresponds to [tex]\( g(x) = 3x^2 \)[/tex]
- [tex]\(-0.333333\)[/tex] corresponds to [tex]\( g(x) = -\frac{1}{3}x^2 \)[/tex]
- [tex]\(0.333333\)[/tex] corresponds to [tex]\( g(x) = \frac{1}{3}x^2 \)[/tex]
- [tex]\(-3\)[/tex] corresponds to [tex]\( g(x) = -3x^2 \)[/tex]
Therefore, all the given functions (A, B, C, and D) are transformations of [tex]\( f(x) = x^2 \)[/tex] with valid, corresponding results.
