To analyze the transformation from the line [tex]\( y = x \)[/tex] to the line [tex]\( y = \frac{x}{5} + 9 \)[/tex], let's break down the components of the new line's equation.
1. Slope:
- For the original line [tex]\( y = x \)[/tex], the slope (denoted as [tex]\( m \)[/tex]) is 1.
- For the transformed line [tex]\( y = \frac{x}{5} + 9 \)[/tex], the slope is the coefficient of [tex]\( x \)[/tex], which is [tex]\(\frac{1}{5}\)[/tex].
Therefore, the slope of the new line is [tex]\(\frac{1}{5}\)[/tex].
2. Comparison of slopes:
- A slope of [tex]\(\frac{1}{5}\)[/tex] is less than a slope of 1. A smaller slope means that the line is less steep, or in other words, it is "flatter."
Hence, the new line is flatter.
3. Y-intercept:
- The y-intercept of a line in the equation [tex]\( y = mx + b \)[/tex] is [tex]\( b \)[/tex], which is the constant term.
- For the transformed line [tex]\( y = \frac{x}{5} + 9 \)[/tex], the y-intercept is 9.
In summary, the slope is [tex]\(\frac{1}{5}\)[/tex], and the line is shifted flatter. The y-intercept of the new line is 9.
