Question
Change y = -3x (proportional) to non proportional.
How did it shift?
How did it shift?
Answers
Answered by
GPT-4o mini
To change the equation \( y = -3x \) from a proportional to a non-proportional relationship, we can introduce a constant term (often referred to as a y-intercept) to the equation. In a proportional relationship, the equation is always of the form \( y = kx \), where \( k \) is a constant and the graph passes through the origin (0,0).
To make the equation non-proportional, we can add a constant, such as \( b \), resulting in:
\[
y = -3x + b
\]
Here, \( b \) can be any real number. For example, if we let \( b = 2 \), the new equation would be:
\[
y = -3x + 2
\]
### How the Graph Shifts:
1. **Proportional Case \( y = -3x \)**:
- The graph is a straight line that passes through the origin (0,0).
- The slope is -3, meaning for every 1 unit increase in \( x \), \( y \) decreases by 3 units.
2. **Non-Proportional Case \( y = -3x + 2 \)**:
- The graph is still a straight line but does not pass through the origin anymore.
- In this case, the y-intercept is 2, which means the line intersects the y-axis at the point (0,2).
- The slope remains -3, indicating the same steepness in the decline, but the entire line shifts up by 2 units.
The addition of the constant term results in a vertical shift of the graph, changing its position while keeping the same slope. In general, non-proportional relationships will have intercepts and will not pass through the origin unless the constant is zero.
To make the equation non-proportional, we can add a constant, such as \( b \), resulting in:
\[
y = -3x + b
\]
Here, \( b \) can be any real number. For example, if we let \( b = 2 \), the new equation would be:
\[
y = -3x + 2
\]
### How the Graph Shifts:
1. **Proportional Case \( y = -3x \)**:
- The graph is a straight line that passes through the origin (0,0).
- The slope is -3, meaning for every 1 unit increase in \( x \), \( y \) decreases by 3 units.
2. **Non-Proportional Case \( y = -3x + 2 \)**:
- The graph is still a straight line but does not pass through the origin anymore.
- In this case, the y-intercept is 2, which means the line intersects the y-axis at the point (0,2).
- The slope remains -3, indicating the same steepness in the decline, but the entire line shifts up by 2 units.
The addition of the constant term results in a vertical shift of the graph, changing its position while keeping the same slope. In general, non-proportional relationships will have intercepts and will not pass through the origin unless the constant is zero.