Question
Use Equation A and describe the differences to show the change from f(x) =x to the equation A.
Equation A: f(x) =3x-4
Equation A: f(x) =3x-4
Answers
GPT-4o mini
To analyze the differences between the functions \( f(x) = x \) and \( f(x) = 3x - 4 \), let's break down the transformations involved.
1. **Understanding the Base Function**:
- The base function \( f(x) = x \) is a linear function with a slope of 1. It is a straight line that passes through the origin (0,0) at a 45-degree angle to both the x-axis and y-axis.
2. **Transforming to Equation A**:
- The new function \( f(x) = 3x - 4 \) is also a linear function, but it has been modified in two ways:
- **Slope Change**: The coefficient of \( x \) in Equation A is 3, which means the slope has increased from 1 to 3. A larger slope indicates that the line is steeper, so for every unit you move to the right along the x-axis, you move up 3 units on the y-axis (instead of 1 unit as with the base function).
- **Y-Intercept Change**: The constant term in Equation A is -4. This means the line is shifted vertically downwards. The y-intercept of \( f(x) = 3x - 4 \) is at (0, -4). In contrast, the y-intercept of \( f(x) = x \) is at (0, 0).
3. **Graphical Representation**:
- If you were to graph both functions, \( f(x) = x \) would be a straight line going through the origin at a 45-degree angle, while \( f(x) = 3x - 4 \) would be a steeper line crossing the y-axis at -4. The new line would intersect the x-axis at a point higher than the original line because the slope is steeper.
4. **Summary of Changes**:
- **Slope**: Increased from 1 to 3 (making it steeper).
- **Y-Intercept**: Shifted from 0 to -4 (moving it down).
In conclusion, transitioning from \( f(x) = x \) to \( f(x) = 3x - 4 \) involves a change in slope which results in a steeper incline, and a downward shift of the function along the y-axis due to the negative y-intercept.
1. **Understanding the Base Function**:
- The base function \( f(x) = x \) is a linear function with a slope of 1. It is a straight line that passes through the origin (0,0) at a 45-degree angle to both the x-axis and y-axis.
2. **Transforming to Equation A**:
- The new function \( f(x) = 3x - 4 \) is also a linear function, but it has been modified in two ways:
- **Slope Change**: The coefficient of \( x \) in Equation A is 3, which means the slope has increased from 1 to 3. A larger slope indicates that the line is steeper, so for every unit you move to the right along the x-axis, you move up 3 units on the y-axis (instead of 1 unit as with the base function).
- **Y-Intercept Change**: The constant term in Equation A is -4. This means the line is shifted vertically downwards. The y-intercept of \( f(x) = 3x - 4 \) is at (0, -4). In contrast, the y-intercept of \( f(x) = x \) is at (0, 0).
3. **Graphical Representation**:
- If you were to graph both functions, \( f(x) = x \) would be a straight line going through the origin at a 45-degree angle, while \( f(x) = 3x - 4 \) would be a steeper line crossing the y-axis at -4. The new line would intersect the x-axis at a point higher than the original line because the slope is steeper.
4. **Summary of Changes**:
- **Slope**: Increased from 1 to 3 (making it steeper).
- **Y-Intercept**: Shifted from 0 to -4 (moving it down).
In conclusion, transitioning from \( f(x) = x \) to \( f(x) = 3x - 4 \) involves a change in slope which results in a steeper incline, and a downward shift of the function along the y-axis due to the negative y-intercept.