Question
Show how to Graph each of these
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
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
To graph the functions you provided, we can outline the key characteristics and behaviors of each function. Below, I will describe how to graph each pair of functions step by step.
### 1. **Graphing \( f(x) = x^4 \) and \( g(x) = -2x^4 \)**
- **For \( f(x) = x^4 \)**:
- This is a quartic function, which has a U-shaped curve and passes through the origin \((0,0)\).
- The function is symmetrical about the y-axis (even function).
- As \( x \) approaches positive or negative infinity, \( f(x) \) increases towards positive infinity.
- **For \( g(x) = -2x^4 \)**:
- This is also a quartic function, but since it has a negative coefficient, it opens downwards.
- It also passes through the origin \((0,0)\) and is symmetrical about the y-axis.
- As \( x \) approaches positive or negative infinity, \( g(x) \) decreases towards negative infinity.
### 2. **Graphing \( f(x) = x^6 \) and \( g(x) = -\frac{1}{4}x^6 \)**
- **For \( f(x) = x^6 \)**:
- Like \( x^4 \), this function opens upwards and has a U-shape.
- It has a higher degree than \( x^4 \), so it increases more rapidly for larger \( |x| \).
- **For \( g(x) = -\frac{1}{4}x^6 \)**:
- This is again a negative quartic function but with a smaller coefficient in front, so it opens downwards.
- The overall behavior is similar to \( g(x) = -2x^4 \), but it is less steep.
### 3. **Graphing \( f(x) = x^3 \) and \( g(x) = 5x^3 + 1 \)**
- **For \( f(x) = x^3 \)**:
- This cubic function has a point of inflection at the origin.
- It passes through the origin and will rise in the first quadrant and fall in the third quadrant.
- **For \( g(x) = 5x^3 + 1 \)**:
- This function shifts the graph of \( f(x) \) up by 1 unit.
- It also has a similar shape to the cubic function but is steeper because of the coefficient \( 5 \).
### 4. **Graphing \( f(x) = x^4 \) and \( g(x) = \frac{1}{2}x^4 + 5 \)**
- **For \( f(x) = x^4 \)**:
- Described earlier.
- **For \( g(x) = \frac{1}{2}x^4 + 5 \)**:
- This function opens upwards and is a vertical stretch of \( f(x) \).
- The graph is shifted up by 5 units, so the vertex is at \((0, 5)\).
### 5. **Graphing \( f(x) = x^5 \) and \( g(x) = \frac{3}{4}(x+4)^5 \)**
- **For \( f(x) = x^5 \)**:
- This cubic-like function passes through the origin and has a point of inflection there.
- It rises steeply in the first quadrant and falls steeply in the third.
- **For \( g(x) = \frac{3}{4}(x + 4)^5 \)**:
- This is \( f(x) \) shifted to the left by 4 units and vertically stretched \( \frac{3}{4} \).
- The inflection point will be at \((-4, 0)\).
### 6. **Graphing \( f(x) = x^4 \) and \( g(x) = (3x)^4 - 2 \)**
- **For \( f(x) = x^4 \)**:
- Described earlier.
- **For \( g(x) = (3x)^4 - 2 \)**:
- This function involves a horizontal stretch by a factor of \( \frac{1}{3} \) and downward shift by 2.
- This means the function has a vertex at \((0, -2)\).
### Graphing Tools
To visualize these functions:
1. **Use graphing software**: Tools like Desmos, GeoGebra, or graphing calculators can easily plot these functions.
2. **Create a table**: For each function, calculate the values of \( y \) for several key \( x \) values to plot points on graph paper manually.
### Conclusion
Each pair of functions has distinct characteristics due to their differences in degrees, coefficients, and translations. The overall shapes can be visualized collectively to show how transformations affect their graphs.
### 1. **Graphing \( f(x) = x^4 \) and \( g(x) = -2x^4 \)**
- **For \( f(x) = x^4 \)**:
- This is a quartic function, which has a U-shaped curve and passes through the origin \((0,0)\).
- The function is symmetrical about the y-axis (even function).
- As \( x \) approaches positive or negative infinity, \( f(x) \) increases towards positive infinity.
- **For \( g(x) = -2x^4 \)**:
- This is also a quartic function, but since it has a negative coefficient, it opens downwards.
- It also passes through the origin \((0,0)\) and is symmetrical about the y-axis.
- As \( x \) approaches positive or negative infinity, \( g(x) \) decreases towards negative infinity.
### 2. **Graphing \( f(x) = x^6 \) and \( g(x) = -\frac{1}{4}x^6 \)**
- **For \( f(x) = x^6 \)**:
- Like \( x^4 \), this function opens upwards and has a U-shape.
- It has a higher degree than \( x^4 \), so it increases more rapidly for larger \( |x| \).
- **For \( g(x) = -\frac{1}{4}x^6 \)**:
- This is again a negative quartic function but with a smaller coefficient in front, so it opens downwards.
- The overall behavior is similar to \( g(x) = -2x^4 \), but it is less steep.
### 3. **Graphing \( f(x) = x^3 \) and \( g(x) = 5x^3 + 1 \)**
- **For \( f(x) = x^3 \)**:
- This cubic function has a point of inflection at the origin.
- It passes through the origin and will rise in the first quadrant and fall in the third quadrant.
- **For \( g(x) = 5x^3 + 1 \)**:
- This function shifts the graph of \( f(x) \) up by 1 unit.
- It also has a similar shape to the cubic function but is steeper because of the coefficient \( 5 \).
### 4. **Graphing \( f(x) = x^4 \) and \( g(x) = \frac{1}{2}x^4 + 5 \)**
- **For \( f(x) = x^4 \)**:
- Described earlier.
- **For \( g(x) = \frac{1}{2}x^4 + 5 \)**:
- This function opens upwards and is a vertical stretch of \( f(x) \).
- The graph is shifted up by 5 units, so the vertex is at \((0, 5)\).
### 5. **Graphing \( f(x) = x^5 \) and \( g(x) = \frac{3}{4}(x+4)^5 \)**
- **For \( f(x) = x^5 \)**:
- This cubic-like function passes through the origin and has a point of inflection there.
- It rises steeply in the first quadrant and falls steeply in the third.
- **For \( g(x) = \frac{3}{4}(x + 4)^5 \)**:
- This is \( f(x) \) shifted to the left by 4 units and vertically stretched \( \frac{3}{4} \).
- The inflection point will be at \((-4, 0)\).
### 6. **Graphing \( f(x) = x^4 \) and \( g(x) = (3x)^4 - 2 \)**
- **For \( f(x) = x^4 \)**:
- Described earlier.
- **For \( g(x) = (3x)^4 - 2 \)**:
- This function involves a horizontal stretch by a factor of \( \frac{1}{3} \) and downward shift by 2.
- This means the function has a vertex at \((0, -2)\).
### Graphing Tools
To visualize these functions:
1. **Use graphing software**: Tools like Desmos, GeoGebra, or graphing calculators can easily plot these functions.
2. **Create a table**: For each function, calculate the values of \( y \) for several key \( x \) values to plot points on graph paper manually.
### Conclusion
Each pair of functions has distinct characteristics due to their differences in degrees, coefficients, and translations. The overall shapes can be visualized collectively to show how transformations affect their graphs.