Question
Change y = ½ x - 4 (non proportional) to proportional.
How did it shift?
How did it shift?
Answers
Answered by
GPT-4o mini
To change the equation \( y = \frac{1}{2}x - 4 \) into a proportional relationship, we need to remove the non-proportional part, which is the constant term (-4).
**Proportional relationships** have the form \( y = kx \), where \( k \) is a constant, and they pass through the origin (0,0).
### Step 1: Remove the constant term
The original equation is:
\[
y = \frac{1}{2}x - 4
\]
To convert it to a proportional form, we only keep the term involving \( x \):
\[
y = \frac{1}{2}x
\]
### Step 2: Understanding the shift
1. **Original Equation**: The graph of \( y = \frac{1}{2}x - 4 \) is a straight line with a slope of \( \frac{1}{2} \) and a y-intercept at -4. This means that the line crosses the y-axis at the point (0, -4).
2. **Proportional Equation**: The graph of \( y = \frac{1}{2}x \) is also a straight line with the same slope of \( \frac{1}{2} \) but passes through the origin (0, 0), since there is no y-intercept.
### Conclusion:
The shift from \( y = \frac{1}{2}x - 4 \) to \( y = \frac{1}{2}x \) removes the constant term (-4), resulting in the graph shifting **upward** by 4 units. As a result, the new line starts from the origin instead of from (0, -4). Hence, this change represents a translation of the line upwards along the y-axis, effectively altering the y-intercept from -4 to 0.
**Proportional relationships** have the form \( y = kx \), where \( k \) is a constant, and they pass through the origin (0,0).
### Step 1: Remove the constant term
The original equation is:
\[
y = \frac{1}{2}x - 4
\]
To convert it to a proportional form, we only keep the term involving \( x \):
\[
y = \frac{1}{2}x
\]
### Step 2: Understanding the shift
1. **Original Equation**: The graph of \( y = \frac{1}{2}x - 4 \) is a straight line with a slope of \( \frac{1}{2} \) and a y-intercept at -4. This means that the line crosses the y-axis at the point (0, -4).
2. **Proportional Equation**: The graph of \( y = \frac{1}{2}x \) is also a straight line with the same slope of \( \frac{1}{2} \) but passes through the origin (0, 0), since there is no y-intercept.
### Conclusion:
The shift from \( y = \frac{1}{2}x - 4 \) to \( y = \frac{1}{2}x \) removes the constant term (-4), resulting in the graph shifting **upward** by 4 units. As a result, the new line starts from the origin instead of from (0, -4). Hence, this change represents a translation of the line upwards along the y-axis, effectively altering the y-intercept from -4 to 0.