Question
Line [Invalid pasted math] is on the coordinate plane and does not pass through the origin. Line [Invalid pasted math] is dilated with a scale factor of 3 and a center of dilation at the origin to create line [Invalid pasted math]. Describe the effects of the dilation on Line [Invalid pasted math]. If line [Invalid pasted math] was dilated with a scale factor of [Invalid pasted math], how would your answer change?(4 points)
Answers
GPT-4o mini
To describe the effects of dilation on a line in the coordinate plane, let's start by considering the basic properties of dilation.
### Original Line
1. **Original line characteristics**: The original line, which we can refer to as Line A, does not pass through the origin, meaning its equation can be expressed as \(y = mx + b\) where \(b \neq 0\).
### Dilation with a Scale Factor of 3
When Line A is dilated with a scale factor of 3 and a center of dilation at the origin (0, 0), several effects can be noticed:
1. **Coordinates Transformation**: Each point \((x, y)\) on Line A will be transformed to \((3x, 3y)\) on the new line, which we can call Line B.
2. **Slope**: The slope of the line remains unchanged since dilation affects the distances but not the angles or slopes. Thus, the slope remains \(m\) for Line B as well.
3. **Y-Intercept Change**: To see how the y-intercept changes, let's consider Line A's y-intercept at point \((0, b)\). Under dilation, this point becomes \((0, 3b)\). The new line Line B will now have a y-intercept of \(3b\), which is vertically scaled away from the origin.
4. **Overall Effect**: The overall effect of the dilation is that the line retains its slope but stretches vertically away from the origin, leading to a new line (Line B) that is parallel to Line A but with a y-intercept that is three times larger than the original.
### Effect of a Different Scale Factor
If Line A was dilated with a different scale factor \(k\) instead of 3 (where \(k \neq 1\)), the effects would be similar:
1. **Coordinates Transformation**: Each point \((x, y)\) will be transformed to \((kx, ky)\).
2. **Slope**: The slope still remains the same at \(m\).
3. **Y-Intercept Change**: The y-intercept of Line A at \((0, b)\) would be transformed to \((0, kb)\), indicating that the new y-intercept is scaled by the factor \(k\).
4. **Overall Effect**: The new line would thus be parallel to the original line, and the y-intercept would change accordingly to become \(kb\). The line would scale vertically and move away from the origin based on the value of \(k\).
In summary, under dilation with a scale factor \(k\), the line would maintain its slope, but the y-intercept would be scaled by the factor \(k\), resulting in a new line parallel to the original line.
### Original Line
1. **Original line characteristics**: The original line, which we can refer to as Line A, does not pass through the origin, meaning its equation can be expressed as \(y = mx + b\) where \(b \neq 0\).
### Dilation with a Scale Factor of 3
When Line A is dilated with a scale factor of 3 and a center of dilation at the origin (0, 0), several effects can be noticed:
1. **Coordinates Transformation**: Each point \((x, y)\) on Line A will be transformed to \((3x, 3y)\) on the new line, which we can call Line B.
2. **Slope**: The slope of the line remains unchanged since dilation affects the distances but not the angles or slopes. Thus, the slope remains \(m\) for Line B as well.
3. **Y-Intercept Change**: To see how the y-intercept changes, let's consider Line A's y-intercept at point \((0, b)\). Under dilation, this point becomes \((0, 3b)\). The new line Line B will now have a y-intercept of \(3b\), which is vertically scaled away from the origin.
4. **Overall Effect**: The overall effect of the dilation is that the line retains its slope but stretches vertically away from the origin, leading to a new line (Line B) that is parallel to Line A but with a y-intercept that is three times larger than the original.
### Effect of a Different Scale Factor
If Line A was dilated with a different scale factor \(k\) instead of 3 (where \(k \neq 1\)), the effects would be similar:
1. **Coordinates Transformation**: Each point \((x, y)\) will be transformed to \((kx, ky)\).
2. **Slope**: The slope still remains the same at \(m\).
3. **Y-Intercept Change**: The y-intercept of Line A at \((0, b)\) would be transformed to \((0, kb)\), indicating that the new y-intercept is scaled by the factor \(k\).
4. **Overall Effect**: The new line would thus be parallel to the original line, and the y-intercept would change accordingly to become \(kb\). The line would scale vertically and move away from the origin based on the value of \(k\).
In summary, under dilation with a scale factor \(k\), the line would maintain its slope, but the y-intercept would be scaled by the factor \(k\), resulting in a new line parallel to the original line.