Question
how to do slope intercept form in reflections
Answers
GPT-4o mini
To understand slope-intercept form and how it relates to reflections in a coordinate plane, let's first clarify what slope-intercept form is. The slope-intercept form of a linear equation is given by:
\[ y = mx + b \]
where:
- \( m \) is the slope of the line,
- \( b \) is the y-intercept (the point where the line crosses the y-axis).
### Reflections in the Coordinate Plane
When reflecting a line in the coordinate plane, we can analyze how its slope and intercept change based on the line of reflection. Below are several common reflection scenarios:
1. **Reflection Across the x-axis:**
- If you reflect the line \( y = mx + b \) across the x-axis, the new equation becomes:
\[
y = -mx - b
\]
- The slope becomes the negative of the original slope, and the y-intercept is also negated.
2. **Reflection Across the y-axis:**
- If you reflect the line across the y-axis, the new equation will be:
\[
y = -mx + b
\]
- The slope is negated, but the y-intercept remains the same.
3. **Reflection Across the Line \( y = x \):**
- To reflect the line \( y = mx + b \) across the line \( y = x \), you swap \( x \) and \( y \) and then solve for \( y \):
\[
x = my + b \implies my = x - b \implies y = \frac{1}{m}x - \frac{b}{m}
\]
- The slope of the reflected line becomes \( \frac{1}{m} \), and the new y-intercept is \( -\frac{b}{m} \).
### Summary
To reflect a line given in slope-intercept form across various axes:
- **x-axis:** Change the sign of both slope and y-intercept: \( y = -mx - b \).
- **y-axis:** Change the sign of the slope only: \( y = -mx + b \).
- **Line \( y = x \):** Swap \( x \) and \( y \), then solve for \( y \): \( y = \frac{1}{m}x - \frac{b}{m} \).
By applying these transformations, you can derive the slope-intercept form of the reflected lines as needed!
\[ y = mx + b \]
where:
- \( m \) is the slope of the line,
- \( b \) is the y-intercept (the point where the line crosses the y-axis).
### Reflections in the Coordinate Plane
When reflecting a line in the coordinate plane, we can analyze how its slope and intercept change based on the line of reflection. Below are several common reflection scenarios:
1. **Reflection Across the x-axis:**
- If you reflect the line \( y = mx + b \) across the x-axis, the new equation becomes:
\[
y = -mx - b
\]
- The slope becomes the negative of the original slope, and the y-intercept is also negated.
2. **Reflection Across the y-axis:**
- If you reflect the line across the y-axis, the new equation will be:
\[
y = -mx + b
\]
- The slope is negated, but the y-intercept remains the same.
3. **Reflection Across the Line \( y = x \):**
- To reflect the line \( y = mx + b \) across the line \( y = x \), you swap \( x \) and \( y \) and then solve for \( y \):
\[
x = my + b \implies my = x - b \implies y = \frac{1}{m}x - \frac{b}{m}
\]
- The slope of the reflected line becomes \( \frac{1}{m} \), and the new y-intercept is \( -\frac{b}{m} \).
### Summary
To reflect a line given in slope-intercept form across various axes:
- **x-axis:** Change the sign of both slope and y-intercept: \( y = -mx - b \).
- **y-axis:** Change the sign of the slope only: \( y = -mx + b \).
- **Line \( y = x \):** Swap \( x \) and \( y \), then solve for \( y \): \( y = \frac{1}{m}x - \frac{b}{m} \).
By applying these transformations, you can derive the slope-intercept form of the reflected lines as needed!