Question
Describe the transformation of f represented
by g. Then graph each function, and show the graph
f(x)=x⁴, g(x) = -2x⁴
f(x)= x, 6 exponent, g(x) = - 1/4x, 6 exponent
f(x)= x³, g(x)= 5x³+1
f(x) =x⁴, g(x)= 1/2x⁴ + 5
f(x) = x, 5 exponent, g(x)= 3/4 (x+4), 5 exponent
f(x) = x⁴, g(x)= (3x)⁴ - 2
by g. Then graph each function, and show the graph
f(x)=x⁴, g(x) = -2x⁴
f(x)= x, 6 exponent, g(x) = - 1/4x, 6 exponent
f(x)= x³, g(x)= 5x³+1
f(x) =x⁴, g(x)= 1/2x⁴ + 5
f(x) = x, 5 exponent, g(x)= 3/4 (x+4), 5 exponent
f(x) = x⁴, g(x)= (3x)⁴ - 2
Answers
GPT-4o mini
Let's describe the transformation of each function \( f \) represented by \( g \) and discuss how to graph each pair.
### 1. Transformations of \( f(x) = x^4 \) to \( g(x) = -2x^4 \)
**Transformation:**
1. **Vertical Stretch:** The factor \(-2\) indicates a vertical stretch by a factor of 2.
2. **Reflection:** The negative sign indicates a reflection over the x-axis.
**Graph:**
- The graph of \( g(x) \) is a stretched and flipped version of \( f(x) \).
### 2. Transformations of \( f(x) = x^6 \) to \( g(x) = -\frac{1}{4}x^6 \)
**Transformation:**
1. **Vertical Compression:** The factor of \(-\frac{1}{4}\) indicates a vertical compression by a factor of 4.
2. **Reflection:** The negative sign indicates a reflection over the x-axis.
**Graph:**
- The graph of \( g(x) \) is a compressed and flipped version of \( f(x) \).
### 3. Transformations of \( f(x) = x^3 \) to \( g(x) = 5x^3 + 1 \)
**Transformation:**
1. **Vertical Stretch:** The factor of \( 5 \) indicates a vertical stretch by a factor of 5.
2. **Vertical Shift:** The \( +1 \) indicates a vertical shift up by 1 unit.
**Graph:**
- The graph of \( g(x) \) will be stretched compared to \( f(x) \), and then shifted upwards.
### 4. Transformations of \( f(x) = x^4 \) to \( g(x) = \frac{1}{2}x^4 + 5 \)
**Transformation:**
1. **Vertical Compression:** The factor of \(\frac{1}{2}\) indicates a vertical compression by a factor of 2.
2. **Vertical Shift:** The \( +5 \) indicates a vertical shift up by 5 units.
**Graph:**
- The graph of \( g(x) \) will be a compressed version of \( f(x) \) and then shifted upwards.
### 5. Transformations of \( f(x) = x^5 \) to \( g(x) = \frac{3}{4}(x+4)^5 \)
**Transformation:**
1. **Horizontal Shift:** The \( (x+4) \) indicates a shift left by 4 units.
2. **Vertical Compression:** The factor of \(\frac{3}{4}\) indicates a vertical compression by a factor of \(\frac{3}{4}\).
**Graph:**
- The graph of \( g(x) \) will be compressed and shifted left compared to \( f(x) \).
### 6. Transformations of \( f(x) = x^4 \) to \( g(x) = (3x)^4 - 2 \)
**Transformation:**
1. **Horizontal Compression:** The factor of \(3\) inside the function indicates a horizontal compression by a factor of \(\frac{1}{3}\).
2. **Vertical Shift:** The \( -2 \) indicates a vertical shift down by 2 units.
**Graph:**
- The graph of \( g(x) \) will be compressed horizontally and shifted down compared to \( f(x) \).
### Graphs
Since I am unable to create visual graphics directly in this text, you can graph these functions using graphing software or a graphing calculator by inputting the functions as shown.
You would input:
1. \( f(x) = x^4 \) and \( g(x) = -2x^4 \)
2. \( f(x) = x^6 \) and \( g(x) = -\frac{1}{4}x^6 \)
3. \( f(x) = x^3 \) and \( g(x) = 5x^3 + 1 \)
4. \( f(x) = x^4 \) and \( g(x) = \frac{1}{2}x^4 + 5 \)
5. \( f(x) = x^5 \) and \( g(x) = \frac{3}{4}(x+4)^5 \)
6. \( f(x) = x^4 \) and \( g(x) = (3x)^4 - 2 \)
This will provide you with clear graphical representations of each function and its transformation.
### 1. Transformations of \( f(x) = x^4 \) to \( g(x) = -2x^4 \)
**Transformation:**
1. **Vertical Stretch:** The factor \(-2\) indicates a vertical stretch by a factor of 2.
2. **Reflection:** The negative sign indicates a reflection over the x-axis.
**Graph:**
- The graph of \( g(x) \) is a stretched and flipped version of \( f(x) \).
### 2. Transformations of \( f(x) = x^6 \) to \( g(x) = -\frac{1}{4}x^6 \)
**Transformation:**
1. **Vertical Compression:** The factor of \(-\frac{1}{4}\) indicates a vertical compression by a factor of 4.
2. **Reflection:** The negative sign indicates a reflection over the x-axis.
**Graph:**
- The graph of \( g(x) \) is a compressed and flipped version of \( f(x) \).
### 3. Transformations of \( f(x) = x^3 \) to \( g(x) = 5x^3 + 1 \)
**Transformation:**
1. **Vertical Stretch:** The factor of \( 5 \) indicates a vertical stretch by a factor of 5.
2. **Vertical Shift:** The \( +1 \) indicates a vertical shift up by 1 unit.
**Graph:**
- The graph of \( g(x) \) will be stretched compared to \( f(x) \), and then shifted upwards.
### 4. Transformations of \( f(x) = x^4 \) to \( g(x) = \frac{1}{2}x^4 + 5 \)
**Transformation:**
1. **Vertical Compression:** The factor of \(\frac{1}{2}\) indicates a vertical compression by a factor of 2.
2. **Vertical Shift:** The \( +5 \) indicates a vertical shift up by 5 units.
**Graph:**
- The graph of \( g(x) \) will be a compressed version of \( f(x) \) and then shifted upwards.
### 5. Transformations of \( f(x) = x^5 \) to \( g(x) = \frac{3}{4}(x+4)^5 \)
**Transformation:**
1. **Horizontal Shift:** The \( (x+4) \) indicates a shift left by 4 units.
2. **Vertical Compression:** The factor of \(\frac{3}{4}\) indicates a vertical compression by a factor of \(\frac{3}{4}\).
**Graph:**
- The graph of \( g(x) \) will be compressed and shifted left compared to \( f(x) \).
### 6. Transformations of \( f(x) = x^4 \) to \( g(x) = (3x)^4 - 2 \)
**Transformation:**
1. **Horizontal Compression:** The factor of \(3\) inside the function indicates a horizontal compression by a factor of \(\frac{1}{3}\).
2. **Vertical Shift:** The \( -2 \) indicates a vertical shift down by 2 units.
**Graph:**
- The graph of \( g(x) \) will be compressed horizontally and shifted down compared to \( f(x) \).
### Graphs
Since I am unable to create visual graphics directly in this text, you can graph these functions using graphing software or a graphing calculator by inputting the functions as shown.
You would input:
1. \( f(x) = x^4 \) and \( g(x) = -2x^4 \)
2. \( f(x) = x^6 \) and \( g(x) = -\frac{1}{4}x^6 \)
3. \( f(x) = x^3 \) and \( g(x) = 5x^3 + 1 \)
4. \( f(x) = x^4 \) and \( g(x) = \frac{1}{2}x^4 + 5 \)
5. \( f(x) = x^5 \) and \( g(x) = \frac{3}{4}(x+4)^5 \)
6. \( f(x) = x^4 \) and \( g(x) = (3x)^4 - 2 \)
This will provide you with clear graphical representations of each function and its transformation.